Magnetic Field Sensing with Nitrogen- Vacancy Color Centers in Diamond

Transcription

1 Magnetic Field Sensing with Nitrogen- Vacancy Color Centers in Diamond The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Pham, Linh My Magnetic Field Sensing with Nitrogen-Vacancy Color Centers in Diamond. Doctoral dissertation, Harvard University. Citable link Terms of Use This article was downloaded from Harvard University s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at nrs.harvard.edu/urn-3:hul.instrepos:dash.current.terms-ofuse#laa

2 Magnetic Field Sensing with Nitrogen-Vacancy Color Centers in Diamond A dissertation presented by Linh My Pham to The School of Engineering and Applied Sciences in partial fulllment of the requirements for the degree of Doctor of Philosophy in the subject of Applied Physics Harvard University Cambridge, Massachusetts May 2013

3 Linh My Pham All rights reserved.

4 Dissertation Advisors: Ronald Walsworth, Marko Lon ar Linh My Pham Magnetic Field Sensing with Nitrogen-Vacancy Color Centers in Diamond Abstract In recent years, the nitrogen-vacancy (NV) center has emerged as a promising magnetic sensor capable of measuring magnetic elds with high sensitivity and spatial resolution under ambient conditions. This combination of characteristics allows NV magnetometers to probe magnetic structures and systems that were previously inaccessible with alternative magnetic sensing technologies This dissertation presents and discusses a number of the initial eorts to demonstrate and improve NV magnetometry. In particular, a wide-eld CCD based NV magnetic eld imager capable of micron-scale spatial resolution is demonstrated; and magnetic eld alignment, preferential NV orientation, and multipulse dynamical decoupling techniques are explored for enhancing magnetic sensitivity. The further application of dynamical decoupling control sequences as a spectral probe to extract information about the dynamics of the NV spin environment is also discussed; such information may be useful for determining optimal diamond sample parameters for dierent applications. Finally, several proposed and recently demonstrated applications which take advantage of NV magnetometers' sensitivity and spatial resolution at room temperature are presented, with particular focus on bio-magnetic eld imaging. iii

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6 Contents Title Page... i Abstract... iii Table of Contents... v List of Figures... ix Citations to Previously Published Work... xi Acknowledgments... xiii 1 NV Background Introduction NV Physical Structure NV Formation NV incorporation during CVD synthesis Nitrogen ion implantation and annealing N-to-NV conversion NV Electronic Structure Coherent manipulation of NV spin state Continuous-Wave Electron Spin Resonance Spectroscopy Rabi Nutations NV spin dephasing, decoherence, and relaxation T 2 spin dephasing time T 2 spin decoherence time T 1 spin relaxation time NV Magnetometry Introduction DC magnetometry: ESR Description Sensitivity Initial demonstrations DC magnetometry: Ramsey sequence Description v

7 Contents Sensitivity Initial demonstrations AC magnetometry Description Sensitivity Initial demonstration Summary NV Spin Coherence Dependence on Magnetic Field Misalignment Introduction NV 13 C hyperne interactions Results Summary NV Magnetic Field Imaging Introduction Experimental setup Micron-scale magnetic eld variations Wide-eld magnetic eld patterns Summary Preferential NV Orientation Introduction Preferential NV orientation in {100} samples Results Discussion Enhanced AC magnetometry Results Discussion Summary Dynamical Decoupling Background Dynamical decoupling at room temperature Results Dynamical decoupling at low temperature Results Discussion Multipulse magnetometry Sensitivity analysis Results Summary vi

8 Contents 7 Spectral Decomposition Background Spectral decomposition technique Spectral decomposition deconvolution procedure Spectral function of a spin bath Results Sample C Samples F and G Discussion Spin bath suppression Ensemble averaging Summary Outlook Sensitivity of an NV magnetic sensor Measurement contrast Number of photons collected per measurement Coherence Time Spatial resolution of an NV magnetic sensor Discussion of trade-os Applications A NV Magnetic Sensitivity 137 A.1 DCmagnetometry: ESR A.2 DCmagnetometry: Ramsey sequence A.3 ACmagnetometry B Experimental Apparatus 146 B.1 CCD based wide-eld uorescence microscope B.2 Scanning confocal microscope B.3 Low temperature setup C Coherence Time T 2 Theoretical Limit 152 D Diamond Samples 157 Bibliography 161 vii

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10 List of Figures 1.1 Four orientations of the NV center in diamond NV center electronic energy level structure at room temperature Behavior of the NV center under optical excitation ESRspectra of NV ensembles NV Rabi nutations Rabi nutation decay at dierent MW drive frequencies T 2 measurement of a single NV center using a Ramsey pulse sequence T 2 measurement of an NV ensemble using CW-ESR T 2 measurement of an NV ensemble using a spin echo pulse sequence T 1 measurement of an NV ensemble Example of an NV ESRbased DC magnetic eld measurement Example of a Ramsey based DC magnetic eld measurement Example of a spin echo based AC magnetic eld measurement Example spin coherence decay measurements Summary of spin coherence decay as a function of misalignment angle between the static magnetic eld and the NV electronic axis Example spin coherence decay measurements in 12 C Sample C Diagram of notched wire pattern Magnetic eld image produced by a notched wire pattern Diagram of zigzag wire pattern Vector magnetic eld image produced by a zigzag wire pattern Four orientations of the NV center in diamond ESRspectra of Samples A and E showing level of preferential orientation Step-ow CVD growth and NV uorescence striations Microscope images of the surface morphologies of Samples A and E Comparison of Sample A and E measurement contrast Comparison of NV magnetic sensitivity for Samples A and E Coherence-normalized NV spin coherence decay of Samples A and E. 78 ix

11 List of Figures 6.1 Pulse diagrams of several dynamical decoupling sequences Sample F example CPMG-n NV spin coherence decay curves Sample B example CPMG-n NV spin coherence decay curves Sample C example CPMG-n NV spin coherence decay curves Scaling of NV ensemble coherence time with number of CPMG pulses Sample E example CPMG-n NV spin coherence decay curves measured at300kand160k NV ensemble spin coherence time as a function of number of CPMG pulses measured from Sample E over a range of temperatures Pure NV spin coherence time as a function of number of CPMG pulses Signal contrast as a function of the number of CPMG pulses AC magnetometry curves from spin echo and 28-and 54-pulse sequences Magnetic sensitivity as a function of number of pulses Magnetic sensitivity as a function of eld frequency CPMG lter functions Sample F example NV spin coherence decay curves Sample F spectral decomposition results Sample C coherence time scaling with number of CPMG pulses Samples C, F, and G spectral decomposition results Monte Carlo simulation of the 13 C induced Overhauser eect Confocal scan of NV uorescence in Sample C SIMS measurement of N concentration in Sample F Photograph showing a bulk diamond substrate's waveguiding eects Plot of the spatial resolution versus magnetic sensitivity of a variety of recently demonstrated magnetic sensing technologies Wide-eld optical and magnetic images of magnetotactic bacteria A.1 Intensity prole of an NV resonance in the ESR spectrum A.2 Sensitivity of a Ramsey based DC magnetic eld measurement A.3 Sensitivity of a spin echo based AC magnetic eld measurement B.1 CCD based wide-eld uorescence microscope B.2 NV scanning confocal microscope B.3 Photograph of the low temperature NV measurement setup x

12 Citations to Previously Published Work A large portion of Chapter 3 has been published in the following paper: Coherence of Nitrogen-Vacancy Electronic Spin Ensembles in Diamond, P.L. Stanwix, L.M. Pham, J.R. Maze, D. Le Sage, T.K. Yeung, P. Cappellaro, P.R. Hemmer, A. Yacoby, M.D. Lukin, R.L. Walsworth. Phys. Rev. B 82, (R) (2010). A large portion of Chapter 4 has been published in the following paper: Magnetic eld imaging with nitrogen-vacancy ensembles, L. M. Pham, D. Le Sage, P.L. Stanwix, T.K. Yeung, D. Glenn, A. Trifonov, P. Cappellaro, P.R. Hemmer, M.D. Lukin, H. Park, A. Yacoby R.L. Walsworth. New J. Phys. 13, (2011). Chapter 5 has been published with minor changes as Enhanced metrology using preferential orientation of nitrogen-vacancy centers in diamond, L. M. Pham, N. Bar-Gill, D. Le Sage, C. Belthangady, A. Stacey, M. Markham, D. J. Twitchen, M. D. Lukin, R. L. Walsworth. Phys. Rev. B 86, (R) (2012). Chapter 6 combines work published in the following two papers: Enhanced solid-state multispin metrology using dynamical decoupling, L. M. Pham, N. Bar-Gill, C. Belthangady, D. Le Sage, P. Cappellaro, M. D. Lukin, A. Yacoby, R. L. Walsworth. Phys. Rev. B 86, (2012). Solid-state electronic spin coherence time approaching one second, N. Bar-Gill, L. M. Pham, A. Jarmola, D. Budker, R. Walsworth. Nat. Commun. 4, 1743 (2013). Chapter 7 has been slightly modied from results published in the following paper: Suppression of spin-bath dynamics for improved coherence of multi-spinqubit systems, N. Bar-Gill, L. M. Pham, C. Belthangady, D. Le Sage, P. Cappellaro, J. R. Maze, M. D. Lukin, A. Yacoby, R. Walsworth. Nat. Commun. 3, 858 (2012). xi

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14 Acknowledgments I have had the opportunity over the past six years to meet and work with many extraordinary people, and it is only thanks to their guidance, support, and friendship that I've accomplished anything at all in graduate school. First of all, I am incredibly grateful to my advisor, Ronald Walsworth, for his guidance in matters both scientic and professional, for his encouragement, and for his unending enthusiasm in the innumerably diverse research projects we have pursued. I admire him as a scientist, as a leader, and as a genuinely good and fun person to be around. I am also sincerely grateful to the post-docs I've worked with over the years. Paul Stanwix and I built the rst Walsworth Lab NV-diamond setup on a corner of an optics table in the the slowed-and stored-light room, and during the (long, dark) three month period between the rst implementation of the experimental setup and our rst measurement of a Rabi nutation, Paul taught me everything I know about aligning optics, thinking about problems systematically, and keeping a good sense of humor in the face of frustration. David Le Sage has thought up so many useful improvements to our NV measurement techniques over the years; he has taught me again and again the importance of taking the time to step back and ask, How can we make this better? Also, his steadfastness and good humor (in addition, of course, to doing half the work) are the reason I got through the NV magnetic eld imager demonstration and writing my rst paper with any sanity left at all. Nir Bar-Gill and I have worked together closely on four published works (represented in three chapters of this dissertation) and numerous other ongoing projects in the three years we've known each other. He is someone I go to for scientic questions, xiii

15 Acknowledgments career guidance, personal advice, and everything in between. He reminds me to think of the bigger picture both in research and in life. And whether we're trying to gure out unexpectedly long nitrogen spin bath correlation times or troubleshooting an experiment from across the country or pulling a week of 16-hour days telling each other the same ve jokes over and over again, he always, always, always makes me laugh. I count myself truly fortunate to have him as a co-worker, mentor, and friend. In addition to Paul, David, and Nir with whom I've worked most closely, the Walsworth NV subgroup has grown considerably over the past ve years, and I have worked with, learned from, and am grateful to everyone in it: Tsun Yeung, David Glenn, Alexei Trifonov, Chinmay Belthangady, Stephen DeVience, Keigo Arai, Huiliang Zhang, Jean-Christophe Jaskula, Anat Bar-Gill, Erik Bauch, and Maddy Corbett. I'd also like to thank the rest of the Walsworth Group in Team Astro-comb (David Phillips, Chih-Hao Li, Alex Glenday, and Nick Langellier) for providing an outside eye to keep us grounded and helping us with data analysis. I am also grateful to have had the opportunity to collaborate with a very rich and active diamond community both within Harvard and around the world. In particular, I'd like to thank Matthew Markham, Alastair Stacey, and Daniel Twitchen at Element Six; Andrey Jarmola and Dmitry Budker at UC Berkeley; Patrick Maletinsky, Mike Grinolds, Sungkun Hong, and Marc Warner in the Yacoby Group; Birgit Hausmann, Tom Babinec, and Jennifer Choy in the Lon ar Group; Peter Maurer, Yiwen Chu, and Nathalie de Leon in the Lukin Group; Igor Aharonovich and Jonathan Lee in the Hu Group; Paola Cappellaro; and Jeronimo Maze. I am unspeakably grateful to my friends. In particular, my roommates, Tom xiv

16 Acknowledgments Kodger, Ted Feldman, and Tom Babinec, have been with me since we all started the Applied Physics PhD program together six years ago. They have been my support, sounding board, and sympathetic ear through twelve courses, roughly twenty problem set all-nighters, a lab move, a lab switch, and endless hours ofpuzzling over experiments. IfI had to suer through a semester ofalternating Quantum and E&M problem sets with anybody, I'm glad it was these guys. My time in graduate school would not have been the same without Robert Natividad, my kindred spirit in food, fun, and philosophy. He was always game for walking three miles out of the way for good food, taking a day trip to New York City for the sole purpose ofseeing Broadway musicals, and spending days mainlining TV shows with me. No matter how stressful life got, hanging out with Robert always made me feel better. I also wanted to especially thank Alyssa Wilsonshe's the most sympathetic person I know, and it's been fun guring the mysteries of cooking and also The Godfather with herand Renee Willett, who has been one of my best friends for more than halfmy life, even when I can only see her once a year. One ofthe highlights ofgraduate school for me has been the Radclie Choral Society. I joined the choir expecting ve hours a week in which I could focus on something that had nothing to do with research or classes; I got so much more than that. I have been reminded again and again how music can not only be beautiful but indescribably powerful, and to have been a part of that is a gift that I will always cherish. The members ofthis choir are some ofthe most extraordinary and extraordinarily giving women I've ever known, and I'm constantly impressed and humbled by how much they can accomplish. Being involved in RCS has been such an xv

17 Acknowledgments enriching experience, and I am so grateful to the conductors, the accompanists, and to every woman whom I've sung with for making it that way. Finally, I want to thank my family. I want to thank my parents for their love and support, for instilling in me a love of learning and encouraging me to pursue any career I wanted. I want to thank my mother for making sure that I always had access to the best education. I want to thank my father for buying me electronics sets and teaching me how to build a computer when I was a girl. It's been a long road from there to here, and I could not have done any of it without their belief in me. I also want to thank my sister Leanne, my grandparents, my aunts and uncles, and all my cousins. xvi

18 Chapter 1 NV Background 1.1 Introduction The nitrogen-vacancy (NV) color center in diamond has recently emerged as a promising candidate platform for a number of applications in quantum sensing, metrology, and information processing. In particular, the NV system possesses a combination of remarkable properties which make it well-suited for these applications. Chief among these properties is the addressability of the NV electronic spin state, which can be initialized via optical pumping, coherently manipulated by applying microwave (MW) elds and standard electron spin resonance (ESR) techniques, and optically detected from the spin-state-dependent uorescence intensity. The NV electronic spin also exhibits long coherence times (T 2 ), even at room temperature; in fact, at T 2 2 ms [1, 2, 3], the NV center holds the record for longest room temperature coherence time of an electronic spin in a solid. Finally, the multitude of solid-state processing techniques that have been developed over the past fty years 1

19 Chapter 1: NV Background by the semiconductor industry can be adapted for diamond. Standard semiconductor processes such as chemical vapor deposition (CVD), ion implantation, and annealing can all be applied to diamond in order to optimize NV concentration and spin impurity environment for a variety of dierent applications. This dissertation focuses on the application of NV centers to magnetic eld sensing, particularly with NV ensembles, though many of the techniques for improving magnetic sensitivity that are discussed are applicable to single centers as well. The rest of Chapter 1 provides a brief introduction to NV centers, including an overview of the processes by which NV centers may be formed in bulk diamond, the NV electronic structure, the standard techniques for manipulating and detecting the NV spin state, and the NV spin's interaction with its environment. Chapter 2 describes several measurement schemes for using NV centers to detect magnetic elds. The remaining chapters present an initial demonstration of a CCD-based NV magnetic eld imager (Chapter 4), discuss several approaches for improving NV magnetic sensitivity (Chapters 3, 5, 6), and demonstrate a spectral decomposition technique that can be applied to gain a better understanding of NV spin bath dynamics (Chapter 7). A brief summary of the current state of NV magnetic eld sensing technology and a discussion of some promising applications are presented in Chapter NV Physical Structure The NV center is a stable defect in diamond's crystal structure; it consists of a substitutional nitrogen atom and a vacancy on adjacent lattice sites, such that the defect's symmetry axis may be oriented along one of four possible crystallographic 2

20 Chapter 1: NV Background N V V N z NV [111] x y NV [111] N V V N NV [111] NV [111] Figure 1.1: Four orientations of the NV center in diamond. Carbon atoms are depicted in black, nitrogen (N) atoms in blue, and vacancies (V) in white. The NV electronic spin is indicated by green arrows. Four additional orientations are possible by ipping the nitrogen atoms and vacancies in each conguration above; however, orientations with equivalent symmetry axes are spectrally indistinguishable and may therefore be considered in the same NV orientation class. directions (see Figure 1.1). These four NV orientation classes are almost always equally populated; however, preferential NV orientation has been observed in some CVD-grown diamond samples (see Chapter 5). The NV center exists predominantly in one of two charge states: the neutral NV 0 state and the negatively charged NV state. The NV charge state, which is believed to receive its additional electron from a nearby substitutional nitrogen atom [4, 5], is the charge state which possesses the myriad useful properties described in the previous section. As a result, the NV is more well-studied, and it is common in the literature 3

21 Chapter 1: NV Background to drop the charge notation, where NV is understood to refer to the negatively charged state of the defect. As the NV charge state is the main focus of the current work, this convention will be adopted for the remainder of this dissertation. It is important to note, however, that the NV 0 charge state remains an area of research both to obtain further insight into the NV charge state and to explore its suitability as another candidate spin system for various quantum applications [6]. 1.3 NV Formation Most of the diamond material used in industrial and research applications today is synthesized using one of two production methods. The high-pressure hightemperature (HPHT) method was rst demonstrated by Tracy Hall at General Electric in 1954 [7] and, as its name suggests, involves growing diamond from a seed crystal under high pressure (> 50, 000 bar) and high temperature (> 1400 C) conditions, where diamond is the stable allotrope of carbon. HPHT synthesis allows for the mass production of high-quality diamond at relatively low cost and is therefore widely used in industrial applications. More recently, chemical vapor deposition (CVD) has emerged as another viable method for diamond production. In CVD synthesis, source gasses containing carbon are introduced into a growth chamber where they are energized to form a plasma that deposits carbon atoms onto a substrate. Compared to HPHT synthesis, CVD growth occurs at lower pressures ( bar) and temperatures (< 1000 C), where the thermodynamically stable allotrope of carbon is graphite rather than diamond. However, by introducing hydrogen gas (H 2 ) in addition to carbon-providing methane 4

22 Chapter 1: NV Background gas (CH 4 ), any graphite deposits are rapidly and selectively etched, resulting in a net growth of diamond even under the described metastable conditions. Furthermore, CVD synthesis has a number of advantages over HPHTsynthesis; diamonds can be grown over large areas and on dierent substrate materials. Also, the introduction of chemical impurities into the growth chamber can be nely controlled and so therefore can the properties of the diamond produced. These advantages make CVD the main synthesis technique for diamonds used in research applications. One of the most common impurities found in diamondboth synthetic and natural is nitrogen, whether as a single substitutional defect (N s ), as part of a nitrogenvacancy defect, or in larger complexes containing multiple nitrogen atoms and/or vacancies. A fair amount of research has been undertaken in recent years to increase the number of NV centers in diamond without increasing the concentration of other defects. This section briey summarizes several common techniques applied for increasing NV concentration NV incorporation during CVD synthesis Nitrogen impurities may be incorporated into diamond during CVD synthesis by owing nitrogen gas (N 2 ) through the growth chamber. The majority of the resultant impurities are substitutional nitrogen defects (N s ); however a small fraction (typically < 0.5%) form NV centers [8]. The concentration of nitrogen and NV defects that are incorporated into the diamond, as well as the diamond growth rate, growth mode, and surface morphology are determined by a combination of growth conditions: amount of nitrogen gas introduced into the growth chamber, substrate temperature, chamber 5

23 Chapter 1: NV Background pressure, etc. Better understanding of how these growth parameters are interrelated is the goal of much current CVD diamond research [9, 10, 11, 12, 13, 14, 8]. However, despite these open questions, Ohno et al. were recently able to CVD-grow a diamond containing a thin layer ( 2 nm thick) of nitrogen-doped material on a high-purity substrate. By additionally electron-irradiating the sample and annealing, a fraction of the nitrogen defects in the thin layer were converted to NV centers [15]. The geometry of a thin NV-layer at the surface of a high-purity diamond is quite useful for a number of NV applications and, until this demonstration, had only been achieved using nitrogen ion implantation (discussed in the next section). Furthermore, the NV centers in these CVD nitrogen delta-doped layers are reported to exhibit less depth dispersion as well as better and more consistent spin coherence properties than those created via nitrogen ion implantation [15] Nitrogen ion implantation and annealing As mentioned previously, NV centers may also be introduced into diamond by implanting nitrogen ions, most commonly N + and N + 2. These ion species often consist of 15 N isotopes in order to distinguish NV centers formed via ion implantation from native NV centers, which contain the more naturally abundant (99.6%) 14 N isotope. Other implant parameters which may be varied include ion uence and implantation time to determine the total nitrogen dose, implantation energy to determine the depth and straggle of ions in the resulting implant layer, and substrate temperature during implantation. Optimizing these parameters to achieve stable NV centers and high N-to-NV conversion eciency has been the subject of much study in recent years and 6

24 Chapter 1: NV Background remains an area of open research [16, 17, 18, 19, 20]; several key topics of this research area are discussed below. The implanted ion mean depth and straggle are functions of the ion species, implantation energy, and target material. The Stopping and Range of Ions in Matter (SRIM) software [21] is used almost exclusively to simulate ion implantation not only for diamond but also in the greater semiconductor community. However, there is one important caveat to consider when using SRIM: the software does not take into account crystalline structure (i.e., it assumes amorphous target material). As a result, channeling eects in crystals likely result in deeper implants than predicted; in fact, experimental evidence for this has been recently demonstrated in diamond [22, 23]). One suggested way to avoid these channeling eects is to implant with a small incidence angle; however, a systematic study in diamond of the actual implant depth compared to the depth predicted by SRIM simulations has still yet to be performed, due in part to the diculty in determining the absolute distance between an NV center and the diamond surface with nanometer-scale resolution. The bombardment of the diamond substrate with nitrogen ions during the implantation step performs the dual role of introducing nitrogen impurities into the diamond and damaging the crystal structure to form vacancies. Subsequent annealing of the sample above 600 C mobilizes the vacancies, which tend to become trapped adjacent to substitutional nitrogen impurities, thus relieving crystal strain due to the slight dierence in the atomic radius of nitrogen compared to carbon. As with the implantation step, a number of parameters during the anneal may be varied to optimize NV stability and N-to-NV conversion eciency (discussed in more detail in the next 7

25 Chapter 1: NV Background section). Forming NV centers via ion implantation has several advantages over forming NV centers by nitrogen doping during CVD diamond growth. Ion implantation allows for more ne control over the dopant dosage and, until recently, the depth and thickness of the dopant layer as well. Also, despite the recent progress demonstrating a nanometers-thickdelta-doped nitrogen layer in CVD-grown diamond [15], much workremains to make the process as reproducible, tunable in terms of layer thickness and nitrogen dosage, and widely available as ion implantation. Thus for now, ion implantation remains the far more accessible option for achieving thin shallow NV layers in diamond substrates. Furthermore, ion implantation allows NV centers to be positioned laterally with 10 nm precision, which may achieved using a focused ion beam [16] or by masking o the diamond surface [24]. As mentioned in the previous section, there are also several disadvantages to ion implantation; implanted NV centers often exhibit poorer optical and spin properties compared to native NV centers. In particular, NV centers formed via ion implantation tend to be less photo-stable [25, 26, 27, 28], experience more photochromism (i.e., spend more time in the NV 0 charge state) [29, 25, 26, 27, 17, 28], and have shorter spin coherence times [17, 30]. The exact causes of these issues are not denitively known; however, several possible culprits include crystal damage from the implantation, surface eects, and the formation of other defects. Better understanding of the ion implantation process will be necessary for improving the quality of implanted NV centers. 8

26 Chapter 1: NV Background N-to-NV conversion In both nitrogen-doped CVD diamond and high-purity diamond that has been implanted with nitrogen ions, substitutional nitrogen defects far outnumber NV centers. Despite eorts to optimize CVD growth conditions, the N-to-NV conversion eciency achieved in as-grown CVD samples does not typically exceed 0.5% [8]. Efforts to improve N-to-NV conversion in implanted samples by optimizing implantation conditions [16, 31, 32, 18, 19] and annealing recipes [19] have been more successful, with a 50% conversion eciency demonstrated for very high nitrogen implantation energies (> 2 MeV) [16, 18]. However, the NV centers formed by these high-energy implants are located deeper in the diamond ( 1μm) with wider depth straggle ( 100 nm) than is generally desired, and lower energy implants which create NV centers within 100 nm of the surface typically have lower N-to-NV conversion (< 10%) [18]. For most NV applications, the expected optimum N-to-NV conversion eciency is 50%, such that each substitutional nitrogen defect can donate its extra electron to an NV center in order to produce the more desirable negatively charged NV state. The limiting factor in NV yield for both implanted and as-grown CVD diamonds is believed to be the concentration of vacancies available in the crystal; as a result, a variety of processing techniques have been explored in recent years in order to create more vacancies and subsequently more NV centers. One popular approach is irradiation (e.g., electron, proton, and neutron) [33, 34, 35]; Another approach is implantation of additional ion species (e.g., He, Ni, C, Ga) [36, 25, 32, 17]. In particular, bombardment with carbon ions should cause damage to the crystal structure, thus forming vacancies without introducing unwanted impurities [17]. The balance in 9

27 Chapter 1: NV Background applying these various processing techniques lies in creating the proper concentration of vacancies without introducing other impurities into the crystal or forming defects that cannot be annealed out. Regardless of whether the additional vacancies are introduced via irradiation or ion implantation, a subsequent anneal step is necessary in order to relieve crystal damage and to allow the vacancies to diuse to substitutional nitrogen defects and form NV centers. As mentioned previously, there has been some eort in recent years to develop optimized annealing recipes. Vacancies become mobile at 600 C [37] and as recently as ve years ago, a standard annealing recipe consisting of a 2-hour anneal at 800 C was thought to be sucient to optimally create NV centers. However, recent work has indicated that longer anneals may be needed to ush out unnecessary vacancies [38], or higher temperature anneals (e.g., up to 1200 ) may be needed to break up other defects [17]. Developing processing recipes to achieve optimized N-to-NV conversion eciency is a research topic of great import to NV ensemble magnetometry; improving N-to-NV conversion is expected to greatly enhance NV magnetic sensitivity, as discussed in detail in the next chapter. 1.4 NV Electronic Structure This section presents a brief description of the NV center's electronic structure and behavior under optical excitation; for a more thorough discussion, refer to [39]. A simplied diagram of the NV center's energy level structure is depicted in Figure 1.2. There are six electrons associated with the negative charge state of the defect: three unpaired electrons from the three carbons nearest to the vacancy, a pair of electrons 10

28 Chapter 1: NV Background 3 E 0 +1 Δ = 2γB 1 D es 1.41 GHz 1 A 1 ZPL = 637 nm 1043 nm 1 E 3 A Δ = 2γB 1 D gs 2.87 GHz triplet states singlet states Figure 1.2: NV center electronic energy level structure at room temperature. The ground state, 3 A 2, is a spin-triplet with a zero-eld splitting D gs 2.87 GHz (1.2 μev) between the 0 and ±1 states. The excited triplet state, 3 E, has smaller a zero-eld splitting D es 1.41 GHz (0.6 μev) and is separated from the ground state by a 637 nm (1.945 ev) zero-phonon line. The ±1 states experience additional splitting Δ proportional to the projection B of an external magnetic eld along the NV symmetry axis. The singlet states, 1 E and 1 A 1, are separated by a 1043 nm (1.190 ev) zero-phonon line. 11

29 Chapter 1: NV Background from the nitrogen (the other three valence electrons associated with the nitrogen form covalent bonds with the three carbons nearest to the nitrogen), and one extra electron believed to be donated from a nearby substitutional nitrogen defect. It is often simpler, however, to consider the NV center as having two holes rather than six electrons. As shown in Figure 1.2, the NV center's ground state is a spin-triplet with 3 A 2 symmetry (C 3v point group), where the m s =0( 0 ) and m s = ±1 ( ±1 ) states experience a zero-eld splitting (D gs 2.87 GHz) due to spin-spin interactions. Applying a small external magnetic eld B along the NV quantization axis lifts the degeneracy of the ±1 energy levels with a Zeeman shift given by Δ=m s γb, where the NV gyromagnetic ratio γ =gμ B /h=2.8 MHz/G. Optical transitions between the ground ( 3 A 2 ) and excited ( 3 E) triplet states have a characteristic zero-phonon line (ZPL) at 637 nm with a broad phonon-sideband ( nm) at room temperature. These optical transitions are primarily spin conserving; however, there exists a primarily non-radiative alternative decay path which allows for the transfer of the spin population from the ±1 state to the 0 state through the singlet states. The precise details of the singlet statesincluding the exact number of singlet states, their symmetries, and their energieshave been a source of contention for many years. The currently accepted model is that there are two singletsthe lower state with 1 E symmetry and the upper state with 1 A 1 symmetryseparated by a 1043 nm ZPL. The energy spacing between the singlet and triplet states is, at the time of this writing, still unknown. For the purposes of describing the behavior of the NV spin state under optical excitation, it is often sucient to treat the singlet states collectively as a manifold with an eective lifetime. 12

30 Chapter 1: NV Background c d 3 E 532 nm nm e 1 A 1, 1 E 3A 2 a b 0 ±1 triplet states singlet states Figure 1.3: Behavior of the NV center under optical excitation described using an eective ve-level system. The resulting simplied electronic structure consists of ve states, shown in Figure 1.3: the triplet ground 0 and ±1 states ( a and b, respectively), the triplet excited 0 and ±1 states ( c and d, respectively), and the singlet state manifold ( e ). As mentioned previously, under above-band optical excitation, the cycling between the ground and excited states is primarily spin conserving, denoted by the solid red lines from c to a and d to b. However, there exists a non-radiative decay path fromexcited states c and d to the metastable singlet manifold e, which has an eective lifetime 200 ns [40]. The transition rate from d to e is faster than the rate from c to e, though the exact ratio of the transitions is not precisely known and likely dependent upon temperature, strain, and other extrinsic 13

31 Chapter 1: NV Background factors [39]. From the singlet manifold, the NV spin decays with approximately equal probability to either of the two ground states a and b. As a result, continuous optical excitation will eventually pump the NV center into the 0 spin state, with maximum polarization determined by the ratio of decay rates from the spin sub-levels of 3 E( c and d ) to the single manifold e. The degree of polarization reported in the literature most commonly falls in the range 80 95% [41, 42, 30, 43, 44, 40]. Furthermore, since the alternate decay path does not result in the emission of a photon in the uorescence band, the uorescence intensity may be used to determine the spin state. At maximum optical driving (i.e., optical saturation, which corresponds to a cycling frequency 77 MHz), the time scale for optical pumping and subsequently the time scale over which the uorescence intensity is integrated to determine the NV spin state is 300 ns. These key characteristics, which allow the NV center's electronic spin state to be initialized through optical pumping and detected via spin-state-dependent uorescence, in combination with the ability to coherently manipulate the NV spin using conventional electron spin resonance techniques (discussed in the following sections), highlight the suitability of the NV center as a platform for applied magnetometry in a variety of areas, including biology, geology, and novel materials. 1.5 Coherent manipulation of NV spin state This section describes several basic electron spin resonance (ESR) techniques that provide the foundation for coherent manipulation of the NV spin state. 14

32 Chapter 1: NV Background Continuous-Wave Electron Spin Resonance Spectroscopy To experimentally determine the NV spin resonances, simultaneous low intensity laser excitation (typically at 532 nm) and low power microwave (MW) radiation are applied. When the MW is o-resonant from all ground-state spin transitions, the MW driving has no eect and the laser excitation optically pumps the NV into the 0 spin state resulting in maximum uorescence intensity. When the MW is resonant with one of the ground-state transitions, e.g., 0 +1, the spin state cycles between the 0 and +1 states, reducing the uorescence intensity. As a result, the NV resonances can be determined by sweeping the MW frequency and monitoring the NV uorescence intensity. When no magnetic eld is applied, there is only one resonance at the zero-eld splitting 2.87 GHz corresponding to the ground-state 0 ± 1 transition [Figure 1.4(a)]. When a static magnetic eld B 0 is applied along an arbitrary direction, the ESR spectrum of a single NV consists of two resonances: one corresponding to the ground-state 0 1 transition and another corresponding to the groundstate 0 +1 transition. The splitting between the two resonances is determined by the projection of the magnetic eld along the NV symmetry axis. In an ensemble measurement where the NV centers may occupy any of four possible orientation classes (Figure 1.1), the ESR spectrum may contain as many as eight transitions [Figure 1.4(b)]. At low enough laser intensities and MW powers, it is also possible to resolve the splitting due to Fermi contact hyperne interactions between the NV electronic spin and the nuclear spin associated with the N in the NV complex, where I =1for 14 N [Figure 1.4(c)] and I =1/2 for 15 N) [Figure 1.4(d)]. 15

33 Chapter 1: NV Background a c Fluorescence Contrast B 0 = 0 G Fluorescence Contrast N MW Frequency (GHz) MW Frequency (GHz) b Fluorescence Contrast B 0 = 70 G d Fluorescence Contrast N MW Frequency (GHz) MW Frequency (GHz) Figure 1.4: ESR spectra of NV ensembles. (a) ESR spectrum at zero applied static magnetic eld. The small splitting is induced by local non-axial strain at the NV center [45, 1, 46, 47]. (b) ESR spectrum with an applied static magnetic eld of 65 G. The eld is applied along an arbitrary direction, resulting in a dierent eld projection on each of the four possible NV symmetry axes and subsequently four pairs of transitions. (c) Hyperne-resolved ESR spectrum of a 0 +1 transition from an 14 N-containing NV ensemble. (d) Hyperne-resolved ESR spectrum of a 0 +1 transition from an 15 N-containing NV ensemble Rabi Nutations As mentioned previously, applying MW driving resonant with one of the groundstate transitions, e.g., 0 +1, induces coherent population oscillations between the two spin sublevels. The standard pulse sequence for measuring these population oscillations, known in conventional magnetic resonance nomenclature as Rabi nutations, is shown in Figure 1.5(a). First, an optical excitation pulse polarizes the NV spin state to 0. Then, a 16

34 Chapter 1: NV Background a initialization detection optical MW b 1.00 Fluorescence Contrast MW Pulse Length (ns) Figure 1.5: (a) Pulse sequence for measuring Rabi nutations. (b) Typical Rabi curve of an NV ensemble,where the uorescence intensity is normalized against the uorescence intensity measured after optical pumping. resonant MW pulse coherently transfers the NV spin population between,e.g.,the 0 and +1 states,and the resultant NV spin state is subsequently detected by measuring the uorescence intensity which results from a short ( 300 ns at optical saturation) optical excitation pulse. The degree of population transfer is a function of both the length and amplitude of the MW pulse; in most NV experiments,however,varying only the MW pulse length is sucient. An example of a typical Rabi curve is shown in Figure 1.5(b),where the normalized uorescence intensity is plotted as a function of MW pulse length. Maximum uorescence intensity corresponds to the NV spin population occupying the 0 state,whereas minimum uorescence intensity corresponds to the NV spin population occupying the +1 state (in this example). Therefore,by applying a series 17

35 Chapter 1: NV Background of MW pulses of specic lengths, the NV spin state can be coherently manipulated for a vast array of more complicated pulse sequences, as discussed in the following section. It is important to note that Rabi nutations do not continue indenitely. There is decay of the Rabi signal due to inhomogeneous broadening, and particularly, there is often beating due to the detuned driving of one or more hyperne transitions associated with the N nuclear spin of the NV (see Figure 1.6). An in depth discussion of these eects is beyond the scope of this dissertation; however, NV Rabi nutations have been studied in great depth in other works [48]. In general, these eects can be mitigated by strong MW driving such that the Rabi nutation frequency is much larger than hyperne detunings ( 2 MHz) and inhomogeneous broadening (depends on the sample and may vary from 100 khz - 10 MHz). The Rabi nutation frequency Ω is given by the following equation: Ω=γB 1 (1.1) where γ =2.8 MHz/G is the gyromagnetic ratio of the NV electronic spin and B 1 is the component of the MW excitation perpendicular to the N-V symmetry axis. 1.6 NV spin dephasing, decoherence, and relaxation Many proposed NV applications (e.g., quantum sensing, metrology, and information processing) rely on the ability to coherently manipulate the NV center's spin state. However, in addition to these applied MW excitation pulses, the NV spin state also experiences perturbations due to interactions with other spin impurities in the 18

36 Chapter 1: NV Background a 1.01 Fluorescence Contrast f f f MW Frequency (GHz) b Fluorescence Contrast f 1 = GHz c d Fluorescence Contrast Fluorescence Contrast f 2 = GHz f 3 = GHz MW Pulse Length (ns) Figure 1.6: Rabi nutation decay at dierent MW drive frequencies. (a) Hyperne-resolved ESR spectrum of a 0 +1 transition from an 14 N- containing NV ensemble. (b-d) Rabi nutations produced by resonantly driving each hyperne transition with equal B 1 4 G. 19

37 Chapter 1: NV Background environment. The most common impurities encountered are the non-zero nuclear spins of the 13 C isotope, which comprise 1.1% of carbon in natural abundance; the electronic spins associated with nitrogen donors not incorporated into NV centers; and, in special cases (involving isotopic engineering and high N-to-NV conversion ef- ciency), other NV spins. Interactions with these spin impurities can cause dephasing and decoherence of the NV spin, characterized by T 2 and T 2 respectively. The NV spin may also interact with lattice phonons, resulting in spin ips and subsequent population decay (i.e., spin-lattice relaxation) on a time scale characterized by T 1. These spin-ip processes are a strong function of temperature and may also have a non-negligible eect on the NV spin decoherence as discussed in Chapter 6. This section briey describes the major contributions to these characteristic time scales (T 2, T 2, and T 1 ) and how each is measured using the conventional ESR techniques described previously T 2 spin dephasing time In the umbrella eld of magnetic resonance, which includes nuclear magnetic resonance (NMR), electron spin resonance (ESR), and magnetic resonance imaging (MRI), the time scale over which dephasing occurs for the central spin or spin ensemble is characterized as the T 2 time. In general, this dephasing occurs as a result of inhomogeneities in the environment; in the particular case of the NV spin, the dominant inhomogeneities are variations in local spin environment experienced by each NV center as well as temporal uctuations in each NV center's local magnetic eld due to dynamics in the surrounding spin bath. 20

38 Chapter 1: NV Background For a standard sample with a natural 1.1% abundance of 13 C, dipolar hyperne coupling with the non-zero 13 C nuclear spin is the dominant source of NV spin dephasing. The nuclei precess in the presence of magnetic elds which are created by local magnetic impurities or externally applied, producing random-phase time varying elds at each NV spin. Furthermore, each NV spin has a dierent local environment of spins and subsequently experiences a dierent local magnetic eld. For natural isotopic abundance of 13 C (1.1%), the 13 C contribution to the NV spin dephasing rate has been measured experimentally both for single NV centers and ensembles to be Γ13 C 10 6 s 1, corresponding to T 2 1 μs [49, 34, 1, 50, 51]. In samples that have been doped with a sucient concentration of nitrogen, dipolar interactions with nitrogen-donated electrons can also contribute signicantly to NV spin dephasing. This contribution is dependent on the nitrogen concentration n N by Γ N (g s μ B ) 2 n N [52, 34], where small dierences in the g-factors associated with the N and NV electronic spins are ignored. Therefore, a nitrogen concentration of 15 ppm ( cm 3 ) would have a contribution to the NV spin dephasing comparable to that of natural abundance 13 C. A possibility that has been pursued in recent years for extending T 2 past a microsecond has been to isotopically engineer high-purity (low nitrogen) diamond with below natural abundance of 13 C. In a sample containing 99.7% 12 C, a dephasing time of T 2 > 10 μs was measured. [1]. The T 2 is typically measured experimentally in one of two ways, described in detail below. One method is to apply a Ramsey pulse sequence and extract the characteristic time of the free induction decay (FID) curve. A typical Ramsey pulse sequence is depicted in Figure 1.7(a); rst, an optical excitation pulse initializes the 21

39 Chapter 1: NV Background a initialization detection optical b 1.00 MW π 2 τ π 2 T* = 180 ± 30 ns 2 Fluorescence Contrast Free Precession Time τ (μs) Figure 1.7: T 2 measurement of a single NV center using a Ramsey pulse sequence. (a) Ramsey pulse sequence. (b) NV free induction decay curve from which T ns can be extracted. NV spin state to 0. Then, a MW π/2 pulse (typically detuned from resonance by δ) transfers the NV spin population into a superposition of, e.g., the 0 and +1 states. After a time τ during which the spin system is allowed to freely precess (and dephase), a second MW π/2 pulse projects the NV spin back into the 0, +1 basis, and the NV spin-state is subsequently detected by measuring the state-dependent uorescence intensity. Varying the free precession time yields an FID curve, which contains beating from the N hyperne transitions (three in the case of 14 N, two in the case of 15 N). The T 2 is extracted from characteristic time of the decay envelope, as shown in Figure 1.7(b). For an ensemble of NV centers, inhomogeneities stemming from the dierent local spin environments experienced by each NV center often broaden the hyperne transi- 22

40 Chapter 1: NV Background a 1.00 b Fluorescence Contrast FWHM (MHz) γ = 0.70 MHz MW Frequency (GHz) Microwave Power Before Gain (dbm) Figure 1.8: T 2 measurement of an NV ensemble using CW-ESR. (a) ESR spectra taken at dierent MW powers and non-zero staticmagneticeld. (b) Linear extrapolation of the value of the FWHM γ at zero MW power gives a dephasing time T ns. tion frequency components of the Ramsey FID to the extent that it becomes dicult to extract T 2 from the curve. An alternative method for measuring the dephasing time T 2 is to take ESR spectra at low optical power, extract the full-width at half maximum (FWHM) of the resonances at progressively lower MW powers, and use the values to linearly extrapolate the FWHM at zero MW power. In Figure 1.8, a series of ESR spectra taken at dierent MW powers are shown, each t with a linear combination of three Lorentzian functions, from which the FWHM γ was extracted. Extrapolating the value of the FWHM γ at zero MW power gives the dephasing time, which is related to the FWHM by T 2 = 1 πγ [34] T 2 spin decoherence time Whereas T 2 is the characteristic time for inhomogeneous spin dephasing, the T 2 time is the characteristic time for homogeneous spin dephasing, more commonly re- 23

41 Chapter 1: NV Background ferred to as spin decoherence. In particular, while T 2 is sensitive to external inhomogeneities, such as in applied static and MW elds, T 2 is intrinsic to the NV center and its local spin environment and is often referred to as spin-spin relaxation in conventional NMR nomenclature. As with inhomogeneous spin dephasing, the major contributors to NV spin decoherence are interactions with 13 C nuclear spins, nitrogen electron spins, and possibly other NV spins. The NV spin coherence time T 2 is typically measured using a spin echo (also occasionally known as a Hahn echo) pulse sequence, shown in Figure 1.9(a). It is simply an extension on the Ramsey pulse sequence: after the NV spin is prepared in a coherent superposition of, e.g., 0 and +1, local magnetic elds cause a phase accumulation during the rst free precession interval τ/2. However, when the local magnetic eld uctuations are slow on the time scale of the free precession interval τ/2, the MW π pulse inverts the accumulated phase such that the same amount of phase is accumulated over the second free precession interval τ, thus refocusing the net phase and ideally eliminating the net phase shift. The ecacy of the refocusing is dependent on how slow the magnetic eld uctuations are in comparison to the free precession interval; faster uctuations result in less perfect refocusing. Atypical NV spin decoherence curve is shown in Figure 1.9; in this and most other coherence measurements presented in this dissertation, the pulse sequence is applied once with the last π/2 MW pulse along the x axis (m 1 ) and again with the last π/2 MW pulse along the -x axis (m 2 ). The signals are then subtracted and normalized [s = (m 1 m 2 )/(m 1 + m 2 )] in order to reject common-mode noise (e.g., optical excitation laser intensity uctuations). For a more detailed and quantitative description of this 24

42 Chapter 1: NV Background a initialization detection optical τ b Fluorescence Contrast MW π π ( π 2) ( ) x x 2, π ( ) x ( 2) -x T = 397 ± 5 μs Spin Echo Time τ (μs) Figure 1.9: T 2 measurement of an NV ensemble using a spin echo pulse sequence. (a) Spin echo pulse sequence. (b) Typical T 2 decoherence curve of an NV ensemble in a commercially available standard grade CVD diamond sample from Element Six, showing the characteristic collapses and revivals in NV spin coherence that result from 13 C Larmor precession in the presence of a magnetic eld. The NV spin coherence time T 2 is extracted from the decay envelope. pulse sequence, see [53]. We now consider the major contributors to spin decoherence in NV centers. As mentioned in the previous section, the Larmor precession of 13 C nuclei is the dominant source of local magnetic eld uctuations in most natural abundance samples (excepting those with high nitrogen concentrations); however, when the free precession time τ is equal to a multiple number of 13 C Larmor periods, phase contributions from each individual 13 C cancel out, leading to collapses and revivals in the observed decoherence curve [see Figure 1.9(b)]. Thus, the true NV spin decoherence is given by the decay of the envelope of Larmor-resonant peaks in the 13 C-induced revivals, 25

43 Chapter 1: NV Background which in a 13 C-dominated sample arises from dipole-dipole coupling and spin-diusion between the 13 C nuclear spins and has a time scale T 2 600μs. These spin dynamics will be discussed in greater detail in Chapter 3. The other typical dominant sources of NV spin decoherence are paramagnetic spins, such as electron spins associated with nitrogen donors (or possibly other NV centers and defects in the sample). These electron spins uctuate on a time scale given by dipole-dipole interactions with the NV spin, described in the previous section by the equation Γ N (g s μ B ) 2 n N. It is in this way that the nitrogen concentration has an enormous impact on the NV spin coherence time. Higher nitrogen concentration is required for higher NV concentration but results in lower NV coherence time. (For example, a sample with N 20 ppm has nitrogen-limited T 2 5 μs.) Both high NV concentrations and long NV coherence times are desirable for maximizing the sensitivity of an NV-based magnetometer, as discussed in Chapter 2. One technique common in conventional NMR for extending coherence is to apply additional MW π pulses to further decouple the central spin from magnetic impurities in its environment. More detailed discussion of these multipulse dynamic decoupling techniques will be presented in Chapter 6. However, it is useful to note here that while in conventional NMR the spin coherence time T 2 often refers to the intrinsic limit of how much the coherence can be extended using multipulse sequences, in the NV literature the T 2 time often refers specically to the characteristic decoherence time scale as measured using a single π pulse spin echo (i.e., Hahn echo) pulse sequence. In this dissertation, we adopt the latter convention and indicate the coherence time measured from n-pulse dynamical decoupling sequences as T (n) 2. 26

44 Chapter 1: NV Background a initialization detection optical MW ( ) π τ b Normalized Signal (arb) T = 6.0 ± 0.3 ms Spin Relaxation Time t (ms) Figure 1.10: T 1 measurement of an NV ensemble. (a) Pulse sequence for measuring T 1. (b) NV spin-relaxation curve T 1 spin relaxation time NV spins also experience relaxation due to spin-ips caused by phononic interactions with the crystal lattice. As a result, the polarization of the NV spin population eventually decays back to the thermally mixed state that exists before optical pumping, with a characteristic time scale T 1. For NV centers in bulk diamond at room temperature, T 1 is typically 6 ms; however, since spin-lattice relaxation is a phononic process, T 1 is quite sensitive to temperature, as discussed in Chapter 6. The pulse sequence for measuring T 1 in NV spins is shown in Figure 1.10; after an initial optical excitation pulse to polarize the NV spin into the 0 groundstate, either a MW π pulse is appliedsuch that the NV spin is transferredto the, e.g., +1 state (m 1 ) or no MW pulse is appliedandthe NV spin is left in the 0 state (m 2 ). 27

45 Chapter 1: NV Background After a time t, an optical detection pulse is applied and the resulting NV uorescence is collected. The subtraction of the two signals (s = m 2 m 1 ) rejects common-mode noise and gives the polarization of the NV spin population remaining after time t. Varying t yields a spin-relaxation curve, from which the characteristic decay time T 1 can be extracted. 28

46 Chapter 2 NV Magnetometry 2.1 Introduction One of the most promising applications for NV centers is as magnetic sensors, both of internal magnetic elds, such as those caused by other spin impurities within the diamond, as well as of magnetic elds produced by external sources, such as biological cells, electronic circuits, advanced materials (e.g., graphene), etc. In the rst case, better understanding of the interactions between dierent spin species (e.g., electron spins associated with substitutional nitrogen and nuclear spins associated with 13 C, 14 N, and 15 N) in the diamond environment may inform engineering decisions for optimizing the NV center for quantum information processing applications and furthermore may be relevant to other solid-state spin systems (e.g., quantum dots, phosphorus donors in silicon, etc.). In the latter case, the ability to image magnetic eld patterns with high magnetic sensitivity and spatial resolution could prove a useful characterization tool in a number of elds. In particular, an NV magnetic 29

47 Chapter 2: NV Magnetometry sensor could be applied to study biomagnetic systems (e.g., neurons, cardiac cells, and magnetic organs used for navigation) [54], geomagnetism, microelectronic circuits, novel materials (e.g, graphene), and nuclear magnetic resonance from organic material at the diamond surface [55, 56]. As mentioned in the previous chapter, the NVcenter possesses a number of useful properties which make it well-suited for magnetic sensing. First of all, since NVcenters are stable point defects embedded in diamond substrates, they can theoretically be localized on a sub-nanometer length scale; as a result, a single-nvmagnetic sensor can likewise achieve sub-nanometer scale spatial resolution. It is important to note, however, that the stand-o distance between the NVcenter and the magnetic eld source limits the size of the magnetic features that can be resolvedthat is, as the NVcenter is moved farther from the source, the magnetic features become blurred. As a result, the spatial resolution of a single-nvmagnetic sensor is more practically limited to a few nanometers. Secondly, NVspins can exhibit long coherence times at room temperature and can subsequently perform sensitive magnetometry under ambient conditions. Alternative magnetic sensing technologies with comparable spatial resolution (e.g., superconducting quantum interference devices and magnetic resonance force microscopes) are limited to operation under cryogenic temperatures. In addition to the ability to operate under ambient conditions, diamonds (and to a lesser extent, NVcenters) are stable over a wide range of temperatures and pressures, allowing for the possibility of magnetic eld measurements under extreme conditions of temperature, pressure, radiation, etc. An NV-diamond magnetometer may also be implemented using a wide variety of 30

48 Chapter 2: NV Magnetometry geometries and magnetic eld sensing modalities. For example, some possible geometries include incorporating an NV center at the tip of a scanning probe, simultaneously imaging an NV ensemble in a thin layer on a bulk diamond chip over a wide eld of view using a charge-coupled device (CCD) array or similar technology, and raster scanning a diraction-limited optical spot using a confocal microscope. An overview of several popular geometries, including their advantages and disadvantages, is given in Chapter 8. This chapter will cover three major magnetic eld sensing schemes for measuring both DCand ACmagnetic elds, including discussions of the sensitivity limits of each scheme. 2.2 DC magnetometry: ESR Description The simplest method of measuring a magnetic eld with NV centers is to take an ESR spectrum and extract the magnetic eld information from the NV resonance frequencies. As described previously, in the presence of a magnetic eld, the ±1 spin statesand subsequently the 0 ± 1 transition frequenciesof an NV center experience Zeeman splitting proportional to the projection of the magnetic eld along the NV symmetry axis. A known static eld [B 0 in Figure 2.1(a)] can be applied along an arbitrary direction to spectrally distinguish the four possible NV orientation classes from each other in a continuous-wave (CW) ESR spectrum [modeled by the blue curve in Figure 2.1(b)]. Any additional unknown static eld δb results in a net static eld B net with ESR spectrum resonances that are shifted from 31

49 Chapter 2: NV Magnetometry a B net δb b NV C NV A B 0 NV B Fluorescence Contrast (arb) NV B NV D NV C NV A NV D MW Frequency (GHz) Figure 2.1: Example of an NV ESR based DC magnetic eld measurement. (a) Diagram showing the known static eld B 0, the measured net magnetic eld B net, and the extracted unknown static eld δb in relation to the four NV symmetry axes. (b) Modeled ESR spectra of an NV ensemble in the presence of the known magnetic eld B 0 (blue) and the net magnetic eld B net (red). By measuring the resonance shifts between these two spectra, the unknown magnetic eld δb can be extracted. those of the known B 0 spectrum [modeled by the red curve in Figure 2.1(b)]. By extracting the resonance shifts which correspond to at least three of the four possible NV orientation classes, one can reconstruct the vector components of the unknown magnetic eld. In cases where the uorescence of NV centers occupying at least three of the four possible NV orientation classes can be collected at once (e.g., in a bulk measurement or when there are multiple NV centers per confocal volume, scanning tip, or wide-eld magnetic eld imager pixel), the vector information of the unknown magnetic eld can be extracted from a single ESR spectrum. In addition to the CW scheme described in Chapter 1, ESR spectroscopy may also be performed using pulses. In a pulsed-esr measurement, a strong laser excitation pulse rst initializes the NV into the 0 spin state. Then, a very low power MW 32

50 Chapter 2: NV Magnetometry pulse may rotate the NV into, e.g., the +1 spin state if the MW frequency is resonant with the 0 +1 spin transition. Finally a second short laser excitation pulse is applied and the resulting uorescence intensity detected to determine the NV spin state. By sweeping the MW frequency, a spectrum similar to what is measured with the CW-ESR scheme is extracted; however, since in the pulsed-esr scheme, the optical excitation and MW radiation are not applied simultaneously and are therefore not competing, the MW power may be lower than in the CW-ESR scheme, thus reducing MW power broadening. Also, the optical power may be stronger than in the CW-ESR scheme, thus increasing the number of photons collected per measurement (given equivalent measurement times for both schemes). As discussed in the next section, these two factors play an important role in the sensitivity achieved by an ESR based magnetic eld measurement Sensitivity The sensitivity η esr of an NV ESR based DC magnetic eld measurement is given by the following relation (for derivation, see Appendix A.1): tm h Δν η esr = P F gμ B α β, (2.1) where P F is a numerical factor determined by the prole of the NV resonance: P F = e/8ln for a Gaussian lineshape, P F =4/ for a Lorentzian lineshape [57]. More importantly, the magnetic eld sensitivity is dependent on resonance contrast α, number of photons collected per measurement β, resonance linewidth Δν, and measurement duration t m. These parameters are not independent of each other: the resonance contrast α may be increased by increasing the power of 33

51 Chapter 2: NV Magnetometry the microwave excitation at the expense of increasing the linewidth Δν due to power broadening. Likewise, the linewidth Δν may be decreased (down to a limit dened by the NV spin dephasing time T 2)by lowering the laser excitation power to reduce power broadening; however, doing so decreases the number of photons collected per measurement β. Optimizing these parameters to achieve the best DC magnetic eld sensitivity using the ESR method yields [57]: η esr P F 2 gμ B 1 α βt 2 (2.2) Note that the optimum magnetic sensitivity given by Equation 2.2 for NV ESR based DC magnetometry can only be achieved using pulsed-esr techniques; CW-ESR suffers from a trade-o between power-broadening at higher optical powers and fewer photons collected per measurement at lower optical powers [57] Initial demonstrations A number of demonstrations of this ESR DC magnetometry scheme have been published to date; however due to the NV center's particular suitability for measuring magnetic elds with high sensitivity and spatial resolution, this section focuses on three initial experimental implementations which also enabled magnetic eld imaging. In the rst demonstration, Maertz et al. [58] imaged magnetic eld vectors resulting from micron-scale permalloy (Ni 0.8 Fe 0.2 )structures patterned onto the surface of a type Ib HPHT diamond. The magnetic eld pattern at the surface of the diamond was scanned using a confocal microscope geometry, and the NV concentration was suciently high that a confocal volume contained all four possible NV orientation classes. Magnetic eld images as large as 40 μm 40 μm were shown, with

52 Chapter 2: NV Magnetometry nm spatial resolution and 20 μt DC magnetic eld resolution (given 10 minutes of averaging per pixel). Steinert et al. [59] published a paper at almost the same time demonstrating the same ESR based DC magnetic eld sensing scheme in a wide-eld microscope geometry in which the NV uorescence is collected from a large eld of view simultaneously and imaged onto a CCD camera. Gold microwires were fabricated on the surface of a diamond which had been implanted with nitrogen ions to form a shallow layer (mean depth 6.7 ± 2.8 nm) of NV centers in an otherwise high-purity substrate. The DC magnetic eld created by running current through the microwires was measured by imaging the NV uorescence across a scan of MW excitation frequencies to reconstruct an ESR spectrum for every pixel. In this way, a 60 μm 60 μm eld of view ESR scan could be acquired in 75 seconds withdiraction-limited spatial resolution 250 nm and magnetic eld resolution 15 μt. Using Equation 2.1 and experimental parameters presented in the paper (α 3%, β tm photons per second, Δν 10.5 MHz, and a Lorentzian lineshape), the expected optimum sensitivity for this demonstration is η esr 32 μt/ Hz. The spatial resolutions of both the scanning confocal and the wide-eld microscope geometries are optical diraction limited. This limit can be surpassed with a scanning-tip geometry, possible implementations for which include nanodiamonds containing single NV centers axed to AFM tips and AFM tips fabricated out of highpurity diamond containing single NV centers in the tip. Schoenfeld and Harneit [60] demonstrated a rst step toward the former of these possible implementations using a slight modication to the ESR based DC magnetic eld sensing scheme. They 35

53 Chapter 2: NV Magnetometry applied a eld-frequency lock in order to measure in real-time the ESR transition frequencies of single NV centers in nanodiamonds with a magnetic eld sensitivity of 6 μt/ Hz. By scanning the position of several nanodiamonds with respect to a steel wire creating a test static magnetic eld, 30 μm 30 μm images of the magnetic eld vectors were acquired with magnetic resolution 30 μt in 10 minutes. The ESR NV magnetometry demonstrations we have discussed thus far used a CW measurement scheme; however as mentioned in the previous section,better magnetic eld sensitivities 1 μt/ Hz are achievable using pulsed-esr techniques. Dréau et al. [57] demonstrated a pulsed-esr optimum magnetic eld sensitivity 0.3μT/ Hz using a single NV center in a high-purity bulk diamond,roughly an order of magnitude improvement over the 2 μt/ Hz achieved using the CW-ESR technique. The ESR spectroscopy based DC magnetometry scheme is a quick,simple,and robust way to extract magnetic eld vector information; however the technique is limited by the linewidth of the ESR resonance,which is fundamentally set by the inverse of the NV dephasing time T 2. This dephasing time is typically on the order of 1 μs in high-purity samples,though T 2 times on the order of 100 μs havebeen observed in high-purity isotopically engineered samples [1]. 2.3 DC magnetometry: Ramsey sequence Description DC magnetic elds can also be measured using a Ramsey pulse sequence. In this measurement scheme,a known static external magnetic eld is typically applied to 36

54 Chapter 2: NV Magnetometry split the ±1 states for technical simplicity. As described previously, after an optical excitation pulse polarizes the NV spin into the 0 state, a MW π/2 pulse prepares the spin into, e.g., a superposition of the 0 and +1 states. The NV spin freely precesses for a time τ, during which it accumulates a phase φ =2πγBτ, where B is the magnitude of the unknown static eld to be measured and γ =gμ B /h 2.8 MHz/G is the NV electronic spin gyromagnetic ratio. After the free precession interval, a second MW π/2 pulse projects the spin backinto the 0, +1 manifold, from which the phase accumulation can be extracted. Actual measurements can be accomplished in two ways. First, the free precession time τ may be swept, and a Fourier transform may be performed on the resulting Ramsey fringes. Any shift δν in the resulting resonance frequencies from what is expected given the known applied static eld can be attributed to the unknown static eld; therefore B can be extracted using the standard Larmor equation B = δν/γ. Alternatively, consider a xed free precession time τ (Figure 2.2); let's dene a as the number of photons collected in one measurement in the absence of the unknown static eld, b as the number of photons collected in one measurement when there is a phase accumulation of φ = π during the free precession time τ, and an average number of photons collected in one measurement β =(a + b)/2. In the presence of the unknown static eld B, the number of photons collected in a measurement S oscillates between a and b as a function of the phase accumulation during the free precession time [as shown in Figure 2.2(b)]: S = (a + b) 2 + (a b) 2 cos (φ) = (a + b) 2 + (a b) 2 cos (γbτ) (2.3) Using Equation 2.3, the magnitude B of the unknown static eld can be extracted 37

55 Chapter 2: NV Magnetometry a 1.00 τ = 0.49 μs b 1.00 Fluorescence Contrast Fluorescence Contrast Free Precession Time τ (μs) Additional Static Field B (G) Figure 2.2: Example of a Ramsey based DC magnetic eld measurement. (a) Modeled Ramsey FID curve in the presence of known magnetic eld B 0. (b) Example magnetometry curve showing the sinusoidal response of the NV uorescence signal to an additional static eld B. The Ramsey pulse sequence free precession time is xed at τ =0.49 μs, indicated by the red arrow in (a). from the number of photons measured S. The main caveat when using this technique is of course that magnetic elds with magnitudes B + 2πm, where m is an integer, all γτ yield the same number of photons S. However, these types of measurements are still quite useful for measuring relative magnetic elds or eld uctuations Sensitivity The sensitivity η ramsey of an NV magnetometer which uses the Ramsey pulse sequence to measure DC magnetic elds is given by (see Appendix A.2 for derivation): η ramsey 1 1 τ gμ B α β (2.4) As in the DC magnetometry scheme utilizing ESR spectroscopy, the magnetic eld sensitivity is dependent on resonance contrast α, number of photons collected per measurement β, and measurement duration τ. Also as before, these parameters are not independent. The measurement contrast decays as a result of NV spin dephas- 38

56 Chapter 2: NV Magnetometry ing, with characteristic time scale T 2. Thus, increasing the measurement duration τ degrades the contrast α. The optimum magnetic eld sensitivity is achieved when τ T 2: η ramsey 1 gμ B α βt 2 (2.5) Initial demonstrations This Ramsey DC magnetometry scheme has been demonstrated qualitatively by Balasubramanian et al. [1], and its magnetic eld sensitivity has been theoretically analyzed in depth [52, 61, 57]; however, a thorough experimental demonstration of NV magnetometry using the Ramsey pulse sequence has yet to be published. As with the ESR spectroscopy based magnetometry technique, however, the magnetic eld sensitivity of this technique is fundamentally limited by the dephasing time T 2, which in NV centers is typically of the order of several microseconds, though T 2 may reach several hundred microseconds in ultra-pure samples [62]. 2.4 AC magnetometry Description Recall that employing a spin echo sequencethat is, inserting an additional MW π pulse in the center of a Ramsey sequencecan extend the characteristic time scale of the NV signal decay from T 2 dephasing to T 2 decoherence. Since the optimum magnetic eld sensitivity of the Ramsey based technique is limited by T 2, a corresponding improvement in sensitivity may be expected by employing a spin echo pulse 39

57 Chapter 2: NV Magnetometry sequence for magnetometry. However, in the spin echo sequence, any phase accumulation due to an unknown static magnetic eld during the rst free precession time will be echoed out by the MW π pulse along with all other static magnetic inhomogeneities. As a result, the spin echo pulse sequence is insensitive to static magnetic elds and not useful for DC magnetometry. Consider instead an oscillating (AC) magnetic eld. If the spin echo pulse sequence is applied such that the MW pulses coincide with eld nodes [Figure 2.3(a)], the net phase accumulated in each half of the pulse sequence is additive rather than subtractive. As a result, the spin echo pulse sequence is useful for measuring AC magnetic elds, where the choice of the full pulse sequence duration determines the AC frequency to which the magnetometry measurement is sensitive. Actual measurements of AC magnetic elds using this technique are accomplished in a manner similar to the latter technique described in the Ramsey DC magnetometry section. Consider an echo pulse sequence of duration τ, which is sensitive to AC magnetic elds of frequency f ac =1/τ. In the absence of an AC magnetic eld, the NV spins do not accumulate net phase during the free precession intervals and are thus projected back into the 0 state at the end of the pulse sequence: i.e., there is no uorescence signal change. In the presence of an AC magnetic eld of amplitude B and frequency f ac, with the pulse sequence optimally timed such that the MW pulses coincide with nodes in the eld, the NV spins accumulate net phase φ =4γBτ during the free precession intervals (see Appendix A.3 for derivation). This phase accumulation, when projected into the measurement basis at the end of the pulse sequence, results in a uorescence signal change. 40

58 Chapter 2: NV Magnetometry a optical MW τ B ac B b Fluorescence Contrast τ = μs Spin Echo Time τ (μs) c 0.03 Fluorescence Contrast AC Magnetic Field Amplitude B (nt) Figure 2.3: Example of a spin echo based AC magnetic eld measurement. (a) Spin echo pulse sequence showing the optimum timing of the MW pulses with respect to the AC magnetic eld to be measured. (b) Measured spin echo decoherence curve of an NV ensemble. The collapses and revivals are characteristic of 13 C Larmor precession (discussed in Chapter 3). (c) Measured magnetometry curve showing the sinusoidal response of the NV uorescence signal to AC magnetic eld amplitude B. The spin echo time corresponding to the magnetic eld frequency f ac = khz is indicated by the red arrow in (b). 41

59 Chapter 2: NV Magnetometry As in the latter Ramsey DC magnetometry technique, the number of photons collected in a measurement S oscillates as a function of the phase accumulation φ during the free precession time τ between a (corresponding to φ =0) and b (corresponding to φ = π) [as shown in Figure 2.3(c)]: S = (a + b) 2 + (a b) 2 cos (φ) = (a + b) 2 + (a b) 2 cos (4γBτ) (2.6) Using Equation 2.6, the amplitude B of the AC magnetic eld can be extracted from the number of photons measured S. Recall, a caveat when using this technique is that magnetic elds with amplitudes B + πm, where m is an integer, all yield the same 2γτ number of photons S. However, these types of measurements are still quite useful for measuring relative magnetic elds or eld uctuations. Additionally, in natural carbon isotopic abundance samples, which exhibit collapses and revivals in the NV spin coherence due to 13 C Larmor precession, the frequency of the AC magnetic eld to be measured must coincide with a revival [see Figure 2.3(b)]: f ac = 1 τ = CB γ13 0, (2.7) 2l where the 13 C nuclear gyromagnetic ratio γ13 C khz/g, B 0 is the magnitude of the static magnetic eld applied to split the NV ground state spin transitions, and l is an integer. This condition is easily met by adjusting the magnitude B 0 of the applied static eld as necessary. Also note that in sub-optimal situations of randomly uctuating magnetic elds or where the MW pulses do not coincide with magnetic eld nodes, the spin echo technique is still able to sense AC magnetic elds with reduced sensitivity, as long as the MW pulses do not coincide with anti-nodes [63, 64]. 42

60 Chapter 2: NV Magnetometry Sensitivity The sensitivity η se of an NV magnetometer employing a spin echo pulse sequence to measure an AC magnetic eld along the NV axis, where the MW pulses are synchronized with nodes in the magnetic eld for optimal sensitivity, is given by (see Appendix A.3 for derivation): η se π 1 1 τ 2gμ B α β (2.8) Here the time of a measurement is approximated by the full free precession time τ. Similar to the other NV magnetometry schemes discussed thus far, the magnetic eld sensitivity is dependent on resonance contrast α, number of photons collected per measurement β, and measurement duration τ. Again, these parameters are not independent; the measurement contrast decays as a result of NV spin decoherence, with characteristic time scale T 2. Thus, increasing the measurement duration τ degrades the contrast α. The optimum magnetic eld sensitivity for an spin echo measurement is achieved when τ T 2 [52]: η se π 1 2gμ B α (2.9) βt 2 As expected, the magnetic eld sensitivity of a spin echo based AC measurement is limited by T 2 rather than T 2 as in the Ramsey based DC measurement. Typical spin echo coherence times in high-purity samples are on the order of 500 μs, though T 2 times on the order of 2 ms have been observed in high-purity isotopically engineered samples [1]. 43

61 Chapter 2: NV Magnetometry Initial demonstration The rst experimental demonstration of the spin echo based NV magnetometry scheme was performed by Maze et al. [65]:using a confocal microscope to probe a single NV center in a high-purity bulk diamond sample, they measured AC magnetic elds of kilohertz frequencies with sensitivity 30 nt/ Hz. They were also able to achieve an AC magnetic eld sensitivity of 0.5 μt/ Hz using a single NV center in a diamond nanocrystal of diameter 30 nm. Since this rst demonstration, NV magnetometers for measuring AC elds have been implemented in a number of other geometries, and much work has been done to improve the magnetic eld sensitivity of this technique by, e.g., improving measurement contrast [66], improving photon collection [67, 68, 69, 70], and extending coherence times either by using specially engineered materials [1] or by employing more complicated pulse sequences [64, 71, 3]. In particular, the rst demonstration of AC magnetic eld imaging using NV ensembles as well as several techniques for improving the sensitivity of AC magnetic eld measurements using NV centers will be covered in detail in Chapters 4, 6, and 5 of this dissertation. Though the spin echo based AC magnetometry scheme is greater than an order of magnitude more sensitive than the T 2-limited DC magnetometry schemes, there are some limitations to the technique. First, due to the sinusoidal response of the NV uorescence signal to AC magnetic eld amplitude, the absolute magnetic eld value cannot be extracted without a priori knowledge of the system. For many applications, however, relative eld measurements are sucient. It is also important to note that the bandwidth of AC frequencies that can be measured is limited by NV 44

62 Chapter 2: NV Magnetometry spin decoherence to 1/T Summary The magnetic eld sensitivities of the three NV magnetic sensing schemes discussed in this chapter can be simplied to the following relation: η 1 α βt (2.10) where the measurement time T is limited to the dephasing time T 2 for DC eld measurements and the coherence time T 2 for AC eld measurements. Equation 2.10 illustrates how the sensitivity of an NV based magnetic eld sensor can be improved by increasing the measurement contrast α, the number of photons collected per measurement β, and the measurement time T (i.e., the coherence time). The remainder of this dissertation will present and discuss a number of eorts to demonstrate and improve AC magnetometry using NV centers in diamond, with particular emphasis on magnetic eld imaging with NV ensembles in a wide-eld CCD based geometry; however many of the techniques for improving magnetic sensitivity are applicable to single centers and alternative geometries as well. Chapter 3 discusses the dependence of the spin echo T 2 on the misalignment angle between the NV symmetry axis and the static eld B 0. Chapter 4 presents the rst demonstration of a wide-eld CCD based NV magnetic eld imager. Chapter 5 explores improving sensitivity by improving measurement contrast via preferential NV orientation. Chapter 6 explores improving sensitivity by improving the coherence time with multipulse sequences. These multipulse sequences can also be employed to study dierent regimes 45

63 Chapter 2: NV Magnetometry of spin environments, as discussed in Chapter 7. Finally, an overview of the various magnetic sensing geometries and modalities that have been proposed for applying NV magnetometry to various applications will be discussed in Chapter 8. 46

64 Chapter 3 NV Spin Coherence Dependence on Magnetic Field Misalignment 3.1 Introduction As discussed in the previous chapter, there are three major factors which aect NV magnetic eld sensitivity: the measurement contrast, the number of photons collected per measurement, and the measurement time (limited by spin decoherence). One approach for improving the number of photons collected per measurement is to simply probe more NV centers by performing ensemble measurements. Ideally in this case, the magnetic eld sensitivity increases as the square root of the number of NV centers in the ensemble. However, practically, ensemble measurements are typically performed on one NV orientation class with the other three classes contributing background uorescence, thereby degrading the measurement contrast as compared to that achieved by probing a single NV center. This issue is addressed in more 47

65 Chapter 3: NV Spin Coherence Dependence on Magnetic Field Misalignment detail in Chapter 5. In addition, increasing the density of NV centers, is typically accompanied by an increase in the density of residual nitrogen paramagnetic impurities, which can become the dominant source of NV spin decoherence, reducing T 2 below the limit set by 13 C nuclear spins alone [72, 73, 48]. Furthermore, the random distribution of 13 C nuclear spins near each NV center results in a large variation in individual NV spin coherence times, which also can aect ensemble coherence. It is worthwhile to note that large spin ensembles with good coherence times are also important in implementing collective quantum memories [74, 75, 76]. Therefore, a detailed understanding of the decoherence mechanisms aecting NV centersboth single centers and ensemblesaccompanied by the development of control techniques to mitigate decoherence is of great importance for both precision magnetometry and quantum information applications. Recall that the spin decoherence of an NV center is governed by interactions with the surrounding bath of nuclear and paramagnetic spins. In high-purity diamond, decoherence is dominated by 13 C nuclear spins (I =1/2) [77], which are dispersed through the crystal with a natural abundance of 1.1%. NV spin coherence times T 2 > 600 μs have been observed at room temperature for individual NV centers in this type of sample [65, 51]. T 2 can be further increased by isotopically engineering the diamond [51]; for example, single NV T 2 > 1.8 ms has been observed at room temperature in ultrapure diamond with 99.7% 12 C [1]. At the time of this work, only limited studies of spin decoherence in ensembles of NV centers had been previously reported: at room temperature, an NV ensemble T 2 58 μs was measured in CVD diamond [78], whereas at low temperature (< 2 K) and high magnetic eld (> 8 48

66 Chapter 3: NV Spin Coherence Dependence on Magnetic Field Misalignment T), an NV ensemble T μs was measured in type 1b HPHT diamond, which degraded to T 2 < 10 μs for temperatures > 20 K [79]. In this chapter, we present an experimental and theoretical study of the coherence properties of NV spin ensembles in room temperature diamond samples of different paramagnetic nitrogen (and consequently NV) concentrations. For Sample A, containing very low nitrogen density and a natural abundance of 13 C, we nd NV ensemble T 2 in excess of 600 μs, comparable to the best results for single NV center measurements in natural isotopic abundance diamond and an order of magnitude greater than previous room temperature ensemble measurements. For Sample B, containing a slightly higher nitrogen density and a natural abundance of 13 C, we nd NV ensemble T μs. Furthermore, for both samples we nd a sharp decrease in the NV ensemble T 2 with misalignment of the static magnetic eld relative to the symmetry axis of the NV orientation class being measured. This behavior is consistent with our theoretical modeling of an ensemble of NV spins interacting with a 13 C spin bath [77]. 3.2 NV 13 C hyperne interactions In high-purity diamond, the decoherence of each individual NV spin is dominated by hyperne interactions with nearby 13 C nuclear spins. The contact term of this interaction decreases exponentially with separation; after a few lattice sites it is not larger than a few megahertz [80]. Meanwhile, the dipolar term of this interaction decreases as r 3 and is responsible for the collapses and revivals observed in NV spin echo measurements [41]. When the externally applied static magnetic eld is aligned 49

67 Chapter 3: NV Spin Coherence Dependence on Magnetic Field Misalignment with the NV symmetry axis, the dipolar eld contributions of all 13 C nuclei cancel after each 2π Larmor precession period of the 13 C nuclear spins, as can be measured by the spin echo sequence [41, 81, 49]. Over longer time scales (> 600 μs for natural abundance 13 C), weak dipole-dipole interactions between 13 C nuclei and between 13 C and paramagnetic impurities induce uctuations (ip-ops) in the NV 13 C hyperne interaction, leading to NV spin decoherence. The predicted form of this decoherence in the spin echo signal of an individual NV is exp [ βτ p ], where the power p in the exponent is between 3 and 4 and the constant β depends on details of the relative location of nearby 13 C nuclei [77]. However, when the magnetic eld makes a small angle θ with the NV axis, the Larmor precession frequency Ω i of individual 13 C nuclei is modied by the hyperne interaction with the NV center, dependent on θ and the relative NV 13 C position r i : Ω i =Ω 0 + δω(θ, r i ) (3.1) As a consequence, 13 C nucleiparticularly those proximal to the NV centerdo not all precess with the same frequency and thus their modulations of the NV spin coherence do not rephase at the same time in the spin echo sequence, inducing NV spin decoherence of the form exp [ α (θ) bτ 2 ] [77]. Here α (θ) describes the misalignment angle dependence of the imperfect spin echo rephasing, and b is proportional to the square of the hyperne interaction between the NV electronic spin and the nearest 13 C nuclear spin. For small θ, α (θ) θ 2. In diamond samples where the N and NV concentrations are low enough that NV centers do not interact signicantly with each other or with nitrogen paramagnetic impurities, the ensemble spin echo signal can be considered as the average of many 50

68 Chapter 3: NV Spin Coherence Dependence on Magnetic Field Misalignment independent signals from individual NV centers. In this case, the ensemble spin echo signal envelope, E(τ), can be signicantly dierent from measurements of single NV centers [82, 83]because an ensemble contains a broad distribution of spin coherence lifetimes due to variations in the positions of 13 C nuclei with respect to individual NV centers in the ensemble: E(τ) = db exp ( α (θ) bτ 2) f (b). (3.2) Here, f (b) is the probability distribution for the magnitude of the NVproximal 13 C hyperne-interaction-squared in an ensemble. Depending on details of this distribution, the ensemble spin echo signal envelope may exhibit Gaussian ( p =2) or even single exponential (p =1) decay at large times τ [82, 83]. 3.3 Results We used a custom-built, wide-eld uorescence microscope (see Appendix B.1 for details) to measure the coherence properties of large ensembles of NV centers. A uniform static magnetic eld was applied using a permanent magnet trimmed by a three-axis set of Helmholtz coils. We extracted the magnitude B 0 of the applied magnetic eld from the timing of the revivals in each spin echo coherence curve. As stated previously, these revivals occur when there is an integer number of 13 C Larmor precession periods during each half of the spin echo sequence, where the 13 C Larmor frequency is given by ω L = γ13 CB 0 (γ13 C khz/g). The misalignment angle θ of the applied magnetic eld relative to the symmetry axis of a given NV orientation class was calculated using the extracted eld magnitude B 0 and the projection of the 51

69 Chapter 3: NV Spin Coherence Dependence on Magnetic Field Misalignment eld along the NV symmetry axis determined from the measured Zeeman splitting of the NV ground-state electronic spin resonances, which is typically dierent for each of the four possible NV orientation classes. We employed two CVDdiamond samples in our NV ensemble measurements. Sample A is a diamond (3 mm 3mm 0.2 mm) with NV density cm 3 ( 0.03 ppb) uniformly across the sample, as determined by uorescence measurements normalized to single NV uorescence made on each sample using a custom-built confocal microscope (see Appendix B.2 for details). Sample B consists of a pure diamond substrate (3 mm 3mm 0.2 mm) with < 1 ppm of ambient nitrogen impurity and < cm 3 (<.006 ppb) NV density, which was overgrown with a N-doped layer 3 μm thick. The density of NV centers in the thin layer region is cm 3 ( 0.6 ppb). The density of nitrogen impurities in each of the samples is 200 times larger than the NV density, estimated from typical N-to-NV conversion eciency in CVDgrown diamonds. The NV ensemble spin coherence times were measured by detecting from regions with cross-sectional area 30 μm 2 (corresponding to 10 3 NV spins) in each sample. Over macroscopic distances ( 1 mm), each diamond sample was found to have uniform spin coherence properties. With the magnetic eld well aligned with the NV spin axis, we measured in each sample NV ensemble coherence times comparable to previous measurements for single centers: T 2 > 600 μs for Sample A and T μs for Sample B. Our measurements also conrm the predicted [77] sharp decrease of NV T 2 with small misalignment of the magnetic eld from the NV axis. Example spin echo measurements at three representative misalignment angles θ for each sample are shown in Figure

70 Chapter 3: NV Spin Coherence Dependence on Magnetic Field Misalignment a 1 θ = 180 ± 5 b 1 θ = ± 0.4 T 2 = 631 ± 9 μs T 2 = 282 ± 7 μs p = 1.20 ± 0.09 p = 1.08 ± θ = ± θ = 22.7 ± 0.4 T 2 = 313 ± 5 μs T 2 = 123 ± 10 μs Probability p = 1.85 ± 0.08 Probability p = 0.96 ± θ = ± θ = 64.2 ± 0.1 T 2 = 184 ± 8 μs T 2 = 23 ± 4 μs p = 1.04 ± 0.06 p = 0.42 ± Spin Echo Time τ (μs) Spin Echo Time τ (μs) Figure 3.1: Example spin echo measurements of NV ensemble spin decoherence. (a) Sample Awith NV density cm 3 ( 0.03 ppb) and N density cm 3 ( 5 ppb); and (b) Sample B with NV density cm 3 ( 0.6 ppb) and N density cm 3 ( 100 ppb). Clearly seen in this data is a sharp decrease in the ensemble NV coherence lifetime T 2 as the B 0 eld is misaligned from the NV spin axis, and as the paramagnetic impurity level increases. For each data plot, the left vertical axis indicates the measured probability for NVs in the ensemble to be in the 0 state at the end of a spin echo experiment, as a function of the echo time τ ; the black line is a t of exp [ (τ/t 2 ) p ] to the overall spin echo signal envelope, where the exponent p is a t parameter and the long term baseline is taken to be 0.5 (the mixed spin state); and the errors in θ, T 2, and p are given by one-standard-deviation condence intervals of the t. The red line is a t to the observed echo signal modulations, which are collapses and revivals of NV spin coherence induced by precession of the natural abundance 13 C nuclear spin bath in applied, hyperne, and dipolar magnetic elds. 53

71 Chapter 3: NV Spin Coherence Dependence on Magnetic Field Misalignment a T 2 Coherence Time (μs) Sample A Sample B b Decay Exponent p Sample A Sample B Misalignment Angle θ (degrees) Misalignment Angle θ (degrees) Figure 3.2: Summary of spin echo measurements as a function of misalignment angle θ between the static magnetic eld and the NV electronic axis: (a) coherence lifetime T 2 and (b) decay exponent p, determined from ts of the signal envelope to exp [ (τ/ T2 ) p ]. Sample A (lower NV and nitrogen concentration) was measured for θ ranging from 90 to 180 degrees; Sample B (higher NV and nitrogen concentration) was measured for θ from 0 to 180 degrees. The solid curve in (a) is the prediction of the NV ensemble decoherence model, as described in the main text. Figure 3.2 summarizes the results of a larger set of systematic measurements of ensemble NV T 2 as a function of θ. In each of these measurements, the spin echo signal envelope was t to exp [ (τ/t 2 ) p ] to determine the coherence lifetime T 2 [Figure 3.2(a)] and decay exponent p [Figure 3.2(b)]. Good agreement is found between our experimentally determined values for T 2 as a function of θ and theoretical predictions [see solid curve in Figure 3.2(a)], which include the decoherence contributions from proximal 13 C nuclear spins via the form exp [ α (θ) bτ 2 ] [77].Decoherence due to 13 C ip-ops was included phenomenologically by multiplying the simulated NV echo signal by exp [ (τ/t 2 (θ =0)) 4]. The low values for the decay exponent p in ensemble experiments is explained by the large dispersion of the dominant NV 13 C dipole-dipole interaction (see Equation 3.2). The distribution of this interaction squared, f (b), was determined by calculating the eect of 13 C nuclei placed randomly 54

72 Chapter 3: NV Spin Coherence Dependence on Magnetic Field Misalignment in the diamond lattice at natural isotopic abundance. For the dominant NV 13 C dipole-dipole interaction, we nd that this distribution scales as f (b) b 3/2 for 13 C nuclei within about 10 nm of an NV center. Thus, f (b) more closely resembles a Lorentzian than a Gaussian distribution, resulting in spin echo signal envelopes that decay more like a single exponential than a Gaussian at large times (see Figure 3.1). Note that for Sample B, the experimentally determined decay exponent is p 1 even for θ 0 [Figure 3.2(b)], which likely results from interactions between NV spins and the higher density of paramagnetic nitrogen impurities in this sample. Spin echo measurements were also performed on a third sample (Sample C) with NV density cm 2 ( 0.6 ppb) and N density cm 2 ( 1 ppm), which was isotopically engineered to contain 99.99% 12 C. In contrast to the two natural isotopic abundance samples, the Sample C spin coherence curves do not exhibit the collapses and revivals induced by the Larmor precession of 13 C nuclear spins in the presence of magnetic elds. Furthermore, long NV ensemble T 2 250μs was observed even at large misalignment angles between the magnetic eld and the NV axis, as illustrated by the comparable NV ensemble spin coherence curves measured at misalignment angles θ 0 and θ 28 shown in Figure 3.3. Note that in the absence of 13 C revivals, the applied magnetic eld orientation and magnitude can also be estimated from the spin transition frequencies of the four NV orientation classes. 3.4 Summary In summary, we demonstrated experimentally that large ensembles of NV centers in high-purity diamond with a natural abundance of 13 C can have spin coherence 55

73 Chapter 3: NV Spin Coherence Dependence on Magnetic Field Misalignment a Probability 1 θ 0 T 2 = 242 ± 3 μs p = 1.28 ± 0.03 b Probability 1 θ T 2 p 28 = 267 ± 2 μs = 1.29 ± Spin Echo Time τ (μs) Spin Echo Time τ (μs) Figure 3.3: Example spin echo measurements of NV ensemble spin decoherence in isotopically engineered Sample C for a magnetic eld that is (a) well aligned to the NV axis and (b) misaligned from the NV axis. The solid black line is a t of exp [ (τ/t 2 ) p ] to the spin echo signal, where the exponent p is a t parameter and the long term baseline is taken to be 0.5 (the mixed spin state). lifetimes at room temperature that are comparable to the best measured for single NV centers (T 2 > 600 μs). We also found a sharp decrease in NV spin coherence time as the applied magnetic eld is misaligned from the NV axis in samples with a natural abundance of 13 C. This behavior is consistent with the predictions of our model for an ensemble of NV spins coupled via position-dependent hyperne interactions to a 13 C nuclear spin bath, leading to imperfect 13 C spin echo revivals and hence, NV decoherence. In contrast, in a sample isotopically engineered with 99.99% 12 C, comparable NV ensemble spin coherence times were observed even for large magnetic eld misalignment angles. Our results demonstrate the potential of NV ensembles for applications in precision magnetometry, combining long spin coherence times at room temperature with the enhanced photon collection provided by probing many NV centers in the detection volume [52]. 56

74 Chapter 4 NV Magnetic Field Imaging 4.1 Introduction In recent years, NV ensembles have been employed to sense DC magnetic elds, demonstrating both vector magnetometry and 2-dimensional magnetic eld imaging [58, 59, 84, 60]. However, as discussed in Chapter 2, the sensitivity of such DC magnetometry schemes is limited by typical μs-scale NV T 2 dephasing times: the best sensitivity experimentally achieved by recent NV ensemble DC magnetometers is 32 μt/ Hz [59]. A natural extension of these eorts, then, is to apply the spin echo based AC magnetometry technique to ensembles of NV centers. The resulting combination of longer measurement time (limited by T 2 rather than T 2 ) and higher photon collection (from probing multiple NV centers rather than a single one) provide dual improvements to the magnetic sensitivity. Furthermore, as demonstrated in Chapter 3, NV ensemble coherence times are not signicantly degraded from coherence times measured from single NV centers, which has important implications in 57

75 Chapter 4: NV Magnetic Field Imaging the ability to maximize the sensitivity of an NV ensemble based AC magnetometer. In this chapter, we discuss a method for 2Dimaging of magnetic elds using an ensemble of NV centers in a thin layer at the surface of a diamond chip. By exciting NV centers conned to a thin surface layer over a large area in bulk diamond and imaging the resulting uorescence onto a CCDcamera, we demonstrate optical detection of 2DAC magnetic eld patterns over large elds of view (> 100 μm) with sub-micron spatial resolution and magnetic sensitivity 100 nt/ Hz per pixel. 4.2 Experimental setup To demonstrate the imaging capabilities of NV ensembles in diamond, we custom built a wide-eld uorescence microscope (see Appendix B.1 for details). Such a wide-eld microscope geometry has the key advantages of being relatively simpler to construct and having a shorter intrinsic image acquisition time over a large eld of view than a scanning confocal microscope. Vector magnetometry is accomplished by resonantly coupling the microwaves to one of the four possible NV orientation classes and sequentially measuring the magnetic eld projection onto the dierent NV vectors. However, there is an important caveat to consider when addressing several NV orientation classes: recall from Chapter 3 that in diamond samples with low nitrogen content and a natural abundance of 13 C isotopes, NV spin coherence is limited by the 13 C bath and is subsequently strongly dependent on small misalignments between the static magnetic eld and the NV axis. Thus, taking measurements with dierent NV orientation classes often practically requires switching between dierent static magnetic elds, each aligned along 58

76 Chapter 4: NV Magnetic Field Imaging a particular NV symmetry axis. It is also important to note that while diamonds with a higher concentration of NV centers in general also contain a higher concentration of nitrogen paramagnetic impurities, which further limit the T 2 coherence time, these samples also produce a larger uorescence signal and are less sensitive to small misalignments in the static eld. In the development of a practical NV ensemble based magnetic eld imager, design parameters such as the required magnetic sensitivity, alignment of the static eld with the NV axis, and range of AC eld frequencies that can be measured will dictate the desired level of impurity concentration. 4.3 Micron-scale magnetic eld variations Resolving magnetic eld patterns that vary over sub-micron length scales places additional constraints on the diamond material properties beyond optimizing NV density and coherence time. Small eld variations become irresolvable at distances greater than their feature size; therefore, in order to image magnetic elds on a submicron length scale, the NV sensors must be conned to a thin layer on the diamond surface and placed within a sub-micron distance of the magnetic eld source. This thin-layer geometry can be achieved either by shallow implantation of nitrogen ions [31, 16] or by altering the growth conditions of CVD-synthesized diamonds such that there is a thin N-doped NV-rich layer at one surface of a high-purity substrate. As discussed in Chapter 1, at the current state of diamond processing technology, shallow implantation oers ner control over the thickness and impurity concentration of the N-doped layer than CVD growth does. Hence, to demonstrate the ability of the NV ensemble magnetic eld imager to resolve eld variations on the micron scale, 59

77 Chapter 4: NV Magnetic Field Imaging we used a high-purity HPHT diamond (Sample D) that had been implanted with N + 2 molecular ions at 15 kev,creating a shallow NV layer with 4 nm vertical straggle, 11 nm below the surface (as estimated from Monte-Carlo simulations using SRIM [21]). From the implant dose and the approximate N-to-NV conversion eciency,we estimate that the 2D NV layer has an NV surface density of cm 2 and an N density of cm 2,resulting in a paramagnetic nitrogenlimited T 2 30 μs. The magnetic eld pattern to be imaged was produced by a copper microwire containing segments of alternating width such that the current density through the wire is increased in the constricted sections,resulting in micron-scale magnetic eld `hotspots'. This notched wire pattern was fabricated on a glass coverslip using photolithography and secured at against the shallow implant NV layer of the diamond sample with a separation distance of 1 μm (Figure 4.1). The proximity of the copper microwire to the NV sensors introduces a complication,however: when the microwave pulses are applied to coherently manipulate the NV spin states,the presence of the metal wire pattern modies the microwave eld prole,resulting in B 1 eld inhomogeneity over the plane of NV sensors. This microwave pickup is not expected to be an issue in the proposed bio-applications for the NV ensemble magnetic eld imager,where the sources of the magnetic eld are,for example,ring neurons rather than metal. We used a permanent magnet to apply a static eld along one of the NV axes perpendicular to the direction of current ow (Figure 4.1) and ran khz AC current through the notched wire pattern. The uorescence as a function of AC 60

78 Chapter 4: NV Magnetic Field Imaging high-purity diamond substrate implanted NV layer I ac Figure 4.1: Diagram of the notched wire structure with respect to the four NV symmetry axes. The magnetic eld measurements were performed on the NV A orientation class. magnetic eld amplitude was calibrated on a pixel-by-pixel basis by sweeping the AC current in order to account for spatial variations in the laser intensity and number of NV centers probed. In principle, such a calibration can also be accomplished using a homogeneous external AC eld before measuring the eld source of interest. Applying microwave pulses resonant with only one NV orientation class, we extracted from the NV uorescence intensity an image of the AC magnetic eld projection along the chosen NV axis. Furthermore, the measured magnetic eld map agrees very closely with our modeled eld map (Figure 4.2). Discrepancies can be attributed to the position and orientation of the wire pattern with respect to the crystallographic structure of the diamond sample being slightly dierent from the estimated values used in the model. We suspect that a small drift of the sample mount position over the course of the measurement is responsible for the slight excess blurring in the 61

79 Chapter 4: NV Magnetic Field Imaging a Modeled 5 μm b Measured 5 μm Magnetic Field Projection (μt) Figure 4.2: (a) Modeled and (b) measured map of the magnetic eld projection along the NV axis produced by running an AC current of frequency khz and amplitude 50 μa through the notched wire pattern at a stand-o distance 1 μm. measured magnetic eld image. The magnetic sensitivity can be measured experimentally as η δb T = δs T/ S B, where δs is the standard error in a set of uorescence signal measurements, and S B is the slope of the uorescence signal S as a function of the AC magnetic eld amplitude B [65]. To determine the sensitivity of the NV magnetic eld imager under the measurement conditions given above, we applied a uniform khz AC eld across the region of interest and measured the standard error δs of the uorescence signal at each pixel. The measurements were conducted at an applied AC eld amplitude corresponding to a point of maximum slope S B in order to optimize the sensitivity, and the average-best sensitivity was calculated as the mean single-pixel sensitivity in a25μm 25 μm area at the center of the gaussian laser excitation prole. Taking into account the time T required to make a magnetic eld measurement, the resulting average-best sensitivity per 614 nm 614 nm pixel was 9 μt/ Hz. Substituting experimental parameters such as measurement contrast, average number of photons 62

80 Chapter 4: NV Magnetic Field Imaging collected per measurement, and measurement time, we calculated the expected shotnoise limited sensitivity per pixel to be 6 μt/ Hz. The measured sensitivity is therefore in reasonable agreement with the shot-noise limited expected value [52]. We attribute the discrepancy to technical noise from the laser, which can be actively stabilized for future work. 4.4 Wide-eld magnetic eld patterns The NV magnetic eld imager is also capable of measuring magnetic eld variations over a wide eld of view simultaneously. For this demonstration, the magnetic eld pattern was generated by counter-propagating AC currents through a pair of copper wires in a zigzag pattern with 100 μm features. The eld was measured by an ensemble of NV sensors occupying a thin layer near the surface of a diamond sample held at against the copper strips (Figure 4.3). Since the magnetic eld pattern in this demonstration does not exhibit large variations in eld at a sub-micron level, it was not necessary to use the nitrogen ionimplanted diamond sample as above. Instead, we employed a CVD-grown sample (Sample B) in which the bulk of the chip was grown in a low-nitrogen environment to produce a high-purity substrate. Additional nitrogen impurities were introduced near the end of the growth process, producing a 3 μm thick layer of N-doped, NV-rich diamond at the surface of the otherwise pure substrate. This layered sample geometry was conrmed by measuring the NV uorescence as a function of depth using a scanning confocal microscope (see Appendix B.2 for details). Recall from Chapter 3 that the N-doped layer of Sample B has an estimated NV density of cm 3 63

81 Chapter 4: NV Magnetic Field Imaging high-purity diamond substrate NV layer z I ac y I ac x Figure 4.3: Diagram of zigzag wire pattern orientation with respect to the four NV axes. ( 0.6 ppb) and an N density of cm 3 ( 100 ppb),resulting in a paramagnetic nitrogen limited T μs. The order of magnitude improvement in coherence time allows for more sensitive magnetometry; hence,such a CVD-grown sample is preferable in situations where the magnetic eld varies on the scale of several microns. The wire pattern was positioned with respect to the diamond chip such that its plane of symmetry coincided with the plane containing the NV axes `NV B' and `NV D' (Figure 4.3). We rst used a permanent magnet to apply a static eld along the `NV A' axis,resonantly coupled the microwave driving eld to that NV population, and ran 4.75 khz AC current through the wire pattern. Following the same measurement procedure described in the previous section,we extracted a 2D map of the resultant AC magnetic eld projection along that axis,as sensed by the ensemble of NV centers near the surface of the diamond [Figure 4.4(a)]. Next,applying a static 64

82 Chapter 4: NV Magnetic Field Imaging a b magnetic field projection (nt) NV A axis 20 μm NV B axis 20 μm 100 c 20 μm magnetic field magnitude (nt) Figure 4.4: Map of the eld projection along (b) NV A and (c) NV B axes measured when an AC current of frequency 4.75 khz and amplitude 50 μa is run through the zigzag wire pattern. (d) A vector eld map of the magnetic eld reconstructed fromeld projections measured along two NV axes. eld of equal magnitude along the `NV B' axis produced a 2D map of the magnetic eld projection along a second axis [Figure 4.4(b)]. Taking advantage of the twofold rotational symmetry of the diamond-wire conguration, we were able to use the two maps to reconstruct the magnetic eld vector in the plane occupied by the NV ensemble [Figure 4.4(c)]. To determine the sensitivity of the NV magnetic eld imager under the above 65

83 Chapter 4: NV Magnetic Field Imaging imaging conditions, we applied a uniform 4.75 khz AC eld across the region of interest and measured the standard deviation δs of the uorescence signal at each pixel. Again, the measurements were conducted at an applied eld amplitude B that corresponded to a point of maximum slope S B in order to optimize the sensitivity. The resulting average-best sensitivity per 614 nm 614 nm pixel was 136 nt/ Hz. Substituting experimental parameters such as measurement contrast, average number of photons collected per measurement, and measurement time, we calculated an expected shot-noise limited sensitivity per pixel 86 nt/ Hz. The measured sensitivity is once again in reasonable agreement with the shot-noise limited expected value with the discrepancy attributable to technical noise from the laser. 4.5 Summary In this work, we demonstrated an NV ensemble magnetic eld imager capable of simultaneously measuring spatially varying AC magnetic eld patterns over large elds of view ( 100 μm) with sub-micron spatial resolution and achieving spin echo based magnetic eld sensitivity 100 nt μm 3/2 / Hz. 1 The preliminary results presented here are an important rst step toward potential biological and materials science applications of the NV ensemble magnetic eld imager. However, we anticipate that a number of technical advances will be required for this device to reach its full potential. In the next several chapters, we will demonstrate some of these improvements. 1 Note that the magnetic eld sensitivity of the CCD-based NV ensemble magnetic eld imager described in [59] was not measured experimentally and a spin echo technique was not employed (S. Steinert, private communication); hence the magnetic eld sensitivity was limited by NV T ns. 66

84 Chapter 5 Preferential NV Orientation 5.1 Introduction In this chapter, we explore the enhancement of NV magnetic eld sensitivity through increased measurement contrast. Specically, we demonstrate preferential orientation of NV centers along two of four possible crystallographic axes in diamonds grown by CVD on the {100} face. This preferential orientation provides increased NV ensemble measurement contrast and enhances AC magnetic eld sensitivity, representing an important step towards NV ensemble based precision magnetometry and high-delity multispin-qubit quantum information processing. As discussed in Chapter 1, due to the diamond crystal structure an NV center can be classied by the orientation of its symmetry axis along one of four possible crystallographic axes: [111], [1 1 1], [ 11 1], and [ 1 11] (Figure 5.1). In most diamond samples, NV centers occupy these four orientations equally. To spectrally distinguish a single NV orientation class, a static magnetic eld is applied along the relevant 67

85 Chapter 5: Preferential NV Orientation N V V z N NV [111] x y NV [111] N V N V NV [111] NV [111] Figure 5.1: Four orientations of the NV center in diamond. Carbon atoms are depicted in black, nitrogen (N) atoms in blue, and vacancies (V) in white. The NV electronic spin is indicated by green arrows. Four additional orientations are possible by ipping the nitrogen atoms and vacancies in each conguration above; however, orientations with equivalent symmetry axes are spectrally indistinguishable and may therefore be considered in the same NV orientation class. crystallographic axis, which maximally splits the degeneracy between ±1 sublevels for the desired NV orientation class [85, 86]. This typically limits coherent NV spin manipulation via a resonant MW eld to only one quarter of the NV population, with the rest of the NV centers contributing background uorescence and thereby degrading measurement contrast. Thus, preferential orientation of NV centers along a subset of the four crystallographic axes would benet NV ensemble applications both by employing a greater fraction of the NV center population as well as by reducing background uorescence. 68

86 Chapter 5: Preferential NV Orientation It was recently shown that for synthetic diamond grown via CVD on {110}- oriented substrates, NV centers can be incorporated into the lattice as a unit and thereby found in only two orientations [8].Here we show for the rst time that preferential NV orientation can also be realized for CVD diamond samples grown on {100}-oriented substrates, which are more commonly available and more compatible with increased area bulk production [87], making them more suitable for applications such as bulk magnetometry.we describe the growth regime and likely mechanism leading to preferential NV orientation.we then present experimental demonstrations that diamond with preferential NV orientation exhibits higher measurement contrast and as a result provides improved AC magnetic eld sensitivity. 5.2 Preferential NV orientation in {100} samples Results We employed a wide-eld uorescence microscope (see Appendix B.1 for details) to perform ESR measurements on ensembles of NV centers in two {100}-oriented CVD-grown bulk diamond samples: one with NV centers populating the four orientation classes equally (Sample A) and one with preferential NV orientation along two crystallographic axes (Sample E).The two samples have similar densities of NV centers [NV cm 3 ( 0.03 ppb) for Sample A, NV cm 3 ( 0.02 ppb) for Sample E] and spin coherence lifetimes (T 2 480μs for Sample A, T 2 530μs for Sample E).Sample A contains a natural isotopic abundance of 13 C (1.1%) whereas Sample E is isotopically pure (0.01% 13 C).Note that though longer NV spin coherence 69

87 Chapter 5: Preferential NV Orientation times have been measured in high-purity diamonds with similar isotopic concentrations of 13 C [1], paramagnetic impurities due to substitutional nitrogen limit the coherence of Sample E to a lifetime that is comparable to that of Sample A. For each sample, NV centers within a 10 μm-diameter cross-sectional area were optically excited and subsequent NV uorescence was collected from a 6 μm 6 μm 50 μm detection volume; the NV spins were driven at Rabi frequency 10 MHz. To determine the relative population of the four NV orientation classes in the two samples, we applied a static eld (B 0 =80G) along each of the four diamond crystallographic axes (i.e., NV axes) and measured the resulting ESR spectra. For such magnetic eld congurations, four NV ESR resonances are observed: one pair of resonances corresponds to the 0 ±1 transitions for the class of on-axis NV centers that are oriented parallel to B 0 (at 2.64 GHz and 3.10 GHz for B 0 =80G); another pair of NV ESR resonances corresponds to the three classes of o-axis NV centers that are not aligned with B 0 (at 2.82 GHz and 2.97 GHz for B 0 =80G). In Sample A, the observed amplitudes of the NV resonances in the ESR spectra were very similar for each of the four static magnetic eld congurations, indicating equal population of NV centers in the four orientation classes [Figure 5.2(a)]. In Sample E, however, the pair of on-axis NV ESR resonances were found to be very weak (i.e., small amplitude) for the static magnetic eld congurations B 0 [ 11 1] and B 0 [ 1 11], indicating a high fraction of NV centers with preferential orientation [Figure 5.2(b)]. From the measured relative amplitudes of the two pairs of NV ESR resonances, we estimate that 94% of the NV centers in the Sample E detection volume are oriented along either the [111] or [1 1 1] directions. Similar ESR spectra measured across Sam- 70

88 Chapter 5: Preferential NV Orientation a B 0 [111] Sample A b B 0 [111] Sample E Fluorescence Contrast B 0 [111] B 0 [111] B 0 [111] Fluorescence Contrast B 0 [111] B 0 [111] B 0 [111] MW Frequency (GHz) MW Frequency (GHz) Figure 5.2: ESR spectra with static magnetic eld (B 0 =80G) along each of four diamond crystallographic axes for (a) Sample A, which exhibits no preferential NV orientation, and (b) Sample E, which exhibits a high level ( 94%) of preferential NV orientation. ple E indicate a comparable fraction of preferential NV orientation (within several percent) over the whole sample surface Discussion In the idealized picture of CVD growth on the (100) surface (or any of the symmetrically equivalent {100} faces), a substitutional nitrogen atom can be incorporated at a lattice site in two congurations: (i) the two remaining bonds above the nitrogen allow a vacancy to form an NV center with a [ 1 11] or [ 11 1] orientation with equal probability or (ii) the two remaining bonds above the nitrogen allow a vacancy to form an NV center with a [ 111] or [ 1 1 1] orientation with equal probability. Thus in the idealized picture of CVD growth on a {100} surface, the four possible NV orientation classes form with equal probability [Figure 5.3(a)]. However, for certain 71

89 Chapter 5: Preferential NV Orientation a (100) plane NV [111] NV [111 ] (011) plane NV [111] NV [111] NV [111] x NV [111] y z b terrace c riser depth (μm) kcounts 60 NV fluorescence striations x (μm) Figure 5.3: (a) Atomic-level model of NV incorporation in (100) step-ow growth,indicating mechanism for preferential NV occupation of only two orientations. (b) Diagram of {100} step-ow growth surface and uorescence striations. (c) Confocal cross-section of a sample which exhibits uorescence striations characteristic of {100} step-ow growth. conditions of substrate temperature and ow of nitrogen gas through the CVD chamber,growth on a {100} surface may occur via a step-ow mode [12]. In this growth mode,the surface morphology is stepped (comprising low angle risers and horizontal terraces),as shown in Figure 5.3(b),which potentially allows the formation of higher angle facets,such as the {110},to form. Furthermore,in the step-ow growth regime NV centers are incorporated more readily into the risers rather than the terraces [88], 72

90 Chapter 5: Preferential NV Orientation a Sample A b Sample E 100 μm 100 μm Figure 5.4: Microscope images of the surface morphologies of (a) Sample A (scanning confocal) and (b) Sample E (dierential interference contrast). which results in visible striations in the uorescence image of the diamond, as shown in Figure 5.3(c). We hypothesize that the preferential incorporation of NV centers in {110} facets on the risers can allow for preferential NV orientation to be realized in {100}-oriented CVD diamond via the same mechanism that allows preferential NV orientation in {110}-oriented CVD diamond [8]. As seen in Figure 5.4(a), the surface morphology of Sample A is not stepped, indicating that it was not grown under step-ow conditions, which is consistent with the observed lack of preferential NV orientation in this sample. In contrast, the surface morphology of Sample E is stepped [Figure 5.4(b)], indicating that it was grown under step-ow conditions, which is consistent with the observed preferential NV orientation of this sample. We have also observed varying levels of preferential NV orientation in several additional {100}-oriented CVD samples exhibiting stepped surface morphology and uorescence striations characteristic of step-ow growth. Detailed characterization of the growth factors determining the presence and level of preferential NV orientation will be the focus of future work. 73

91 Chapter 5: Preferential NV Orientation Fluorescence Contrast Sample A Sample E Spin Echo Time τ (μs) Figure 5.5: Comparison of the measurement contrast of the NV spin coherence decay curves measured for the non-oriented Sample A and preferientallyoriented Sample E. 5.3 Enhanced AC magnetometry Results In Figure 5.5 we present NV spin coherence decay curves that demonstrate the expected factor of 2 improvement in NV ensemble measurement contrast for Sample E (preferential NV orientation along two axes) relative to Sample A (equal populations of all NV orientations). Note that Sample A exhibits collapses and revivals in the measured NV spin coherence, which result from the Larmor precession of 13 C impurities in this sample [41]. This eect does not alter the decoherence envelope, which is similar for both samples. As discussed in Chapter 2, the improvement in measurement contrast for samples with preferential NV orientation improves NV magnetic sensitivity. Recall, in the standard AC magnetometry measurement utilizing a spin echo pulse sequence, the 74

92 Chapter 5: Preferential NV Orientation a 0.06 b 1 Fluorescence Contrast δs δb δb Field Uncertainty δb (nt) 0.1 Sample A Sample E AC Magnetic Field Amplitude B (nt) Measurement Time T (s) Figure 5.6: Comparison of NV magnetic sensitivity for Samples A and E. (a) NV magnetometry signal (i.e.,uorescence contrast) due to an AC magnetic eld of frequency f ac =3.08 khz as a function of the magnetic eld amplitude B,acquired using same method as described in Chapter 4. (b) Uncertainty δb in the measured AC magnetic eld amplitude as a function of measurement time T. As discussed previously,the magnetic sensitivity η se δb T. Note that the measurement contrast of Sample E is 2 times larger than of Sample A,leading to similarly improved magnetic eld sensitivity. optimum AC magnetic eld sensitivity is given approximately by [52]: η se π 1 2gμ B α(t ), (5.1) βτ ac where α is the measurement contrast, β is the number of photons collected per measurement,and the measurement time is approximated by the period of the AC magnetic eld τ ac. Note that the measurement contrast is modied by NV spin decoherence over the measurement time T τ ac. In Figure 5.6(a),we plot the measured NV magnetometry signal (i.e.,uorescence contrast) as a function of applied AC magnetic eld amplitude for both the preferentially oriented Sample E and the standard (non-oriented) Sample A,using the spin echo sequence with an AC magnetic eld of frequency f ac =3.08 khz. The enhanced measurement contrast from the preferentially oriented sample improves the magnetic eld sensitivity by a factor 2. As shown in Figure 5.6(b),we nd that the AC 75

93 Chapter 5: Preferential NV Orientation magnetic eld sensitivity in the detection volume (6 μm 6μ m 50 μm) for the preferentially oriented Sample E is 2.6 ± 0.1nT/ Hz, whereas the sensitivity for the standard Sample A is 6.1 ± 0.2nT/ Hz Discussion Other contributions to AC magnetic eld sensitivity Samples A and E were chosen for this work because they are similar in most aspects excepting the presence or absence of preferential NV orientation. However, the two samples are not identical; they dier slightly in isotopic composition, NV density, and spin coherence time. Therefore, in order to properly compare the eect of preferential NV orientation on AC magnetic eld sensitivity, we must perform a detailed analysis of the contributions from these other dierences. The AC magnetic eld sensitivity given by Equation 5.1 can be re-written as: η se π 1 2gμ B α 0 exp [ (τ ac /T 2 ) p ] 1 (5.2) β0 n NV τ ac The Equation 5.1 parameter α(t ) is here recast in terms of the initial measurement contrast α 0 and the reduction in measurement contrast at measurement time T τ ac due to decoherence characterized by T 2 and exponential power p. The number of NV uorescence photons collected per measurement β depends on the number of NV spins n NV contributing to the measurement and the number of photons collected from a single NV in one measurement β 0, which in turn is dependent on collection eciency, and factors such as excitation laser power and optical detection pulse duration. We now consider the contributions of dierences in sample isotopic composition, NV density, and spin coherence time separately. 76

94 Chapter 5: Preferential NV Orientation Isotopic composition Sample A has a natural abundance (1.1%) of 13 C, whereas Sample E was isotopically engineered to contain 0.01% 13 C. As mentioned previously, the higher concentration of 13 C in Sample A results in periodic collapses and revivals characteristic of 13 C Larmor precession in the coherence curve (Figure 5.5). By performing measurements on an AC magnetic eld with a frequency corresponding to the center of a revival (or alternatively, by applying a static magnetic eld such that the center of a revival corresponds to the frequency of the AC magnetic eld one desires to measure), the collapses and revivals may be ignored, and only the eect of 13 C on the decoherence envelope need be considered. We also note that although the diamond lattice parameters are quantitatively dierent for samples with dierent isotopic concentrations of carbon, the growth morphology for these particular samples is dominated by the growth conditions and especially the amount of nitrogen incorporated. As such, there is no distinguishable eect due to the 12 C diamond lattice parameter on the growth morphology of these samples. Coherence time In Figure 5.7, the NV spin decoherence curves are replotted in terms of coherence rather than contrast. From the ts to the decoherence envelope (Sample A) and curve (Sample E), we extract t parameters T 2 = 479 ± 2 μs and p =2.46 ± 0.04 for Sample A, and T 2 = 527 ± 1 μs and p =2.01 ± 0.02 for Sample E. Even though there are small dierences in the decoherence behavior of the samples, the actual coherence levels at the time τ ac = μs corresponding to the frequency of the applied AC magnetic eld are quite similar: for Sample A and for Sample E, as calculated from the ts. Thus we expect that the dierence in NV spin coherence time between the two samples will contribute to a factor improvement 77

95 Chapter 5: Preferential NV Orientation 1 τ ac = μs Sample A Sample E 0.8 Coherence Spin Echo Time τ (μs) Figure 5.7: Coherence-normalized NV spin coherence decay of Samples A and E, where the thicker lines denote ts to decoherence envelopes of the form exp [ (τ/t 2 ) p ]. The AC magnetic eld period τ ac is indicated with an arrow. in the sensitivity of Sample E compared to that of Sample A. NV density The NV densities of Samples A and E were determined from the NV uorescence measured using a scanning confocal microscope (see Appendix B.2 for details). The NV densities were found to be NV cm 3 ( 0.03 ppb) for Sample A, and NV cm 3 ( 0.02 ppb) for Sample E. However, the sensitivity is dependent on the number of NV spins contributing to the measurement n NV ;thus only one quarter of the NV centers in Sample A contribute (n NV cm 3 ) whereas roughly one half of the NV centers in Sample E contribute (n NV cm 3 ). As a result, we expect that the dierence in NV density between the two samples will contribute to a factor improvement in the sensitivity of Sample E compared to that of Sample A. 78

96 Chapter 5: Preferential NV Orientation Combined eect We measured AC magnetic eld sensitivity (in a 6 μm 6 μm 50 μm detection volume) to be 6.1 ± 0.2nT/ Hz for Sample A and 2.6 ± 0.1nT/ Hz for Sample E, corresponding to a factor 2.3 ± 0.1 dierence in measured sensitivity. The combined eect of the small dierences in spin coherence time and NV density between the two samples gives a factor 1.07 improvement in the sensitivity of Sample E compared to that of Sample A. Removing these contributions from the measured sensitivity still yields a factor > 2 improvement in the sensitivity of the preferentially oriented Sample E compared to the standard Sample A. Limitations of preferential NV orientation Thoughsamples withpreferential NV orientation provide a simple way to improve NV ensemble magnetometry sensitivity by a factor of 2, there are some caveats to keep in mind. First, since magnetic eld projections in three directions are necessary for a full vector reconstruction, samples in which NV centers are preferentially oriented along only two crystallographic axes are less compatible with vector magnetometry. Second, it should be noted that the step-ow growth mode and corresponding uorescence striations indicate inhomogeneity in NV distribution over 10μm. However, this eect is less important for measurements using bulk diamond and may be removed throughcareful calibration in samples withthinner NV layers. Finally, preferential orientation of NV centers has thus far only been observed in CVD-grown samples, as a byproduct of the growth process. Unfortunately, typical N-to-NV conversion achieved in as-grown CVD diamonds is typically < 0.5%, and standard processing techniques to increase the NV concentration (e.g., nitrogen ion implantation, irradiation, and 79

97 Chapter 5: Preferential NV Orientation annealing) also destroy the preferential NV orientation. This limitation may possibly be overcome by introducing external strain to the crystal during annealing to cause the NV centers to re-orient along a preferred direction. Such a technique has been demonstrated with neutral carbon <100>-split interstitial defects in diamond [89]; however, the same has yet to be attempted on NV centers. 5.4 Summary In summary, we demonstrated that CVD diamond grown on the {100} crystallographic face can yield NV centers that are preferentially oriented along two of the four possible crystallographic axes. We attribute this eect to the creation of terraces and risers in the step-ow growth morphology such that NV centers are incorporated mainly into the risers, which can contain higher angle facets such as {110} and thus accommodate only two NV orientations. We showed that this preferential orientation increases the NV ensemble measurement contrast by about a factor of two and enables a similar enhancement in AC magnetometry sensitivity. The discovery of preferential orientation in diamond grown on the {100} surface, which is the most widely available and frequently used type of CVD diamond, could inform future research in material science and allow the design and growth of optimized diamond samples with tailored NV orientations to benet a wide variety of applications in precision magnetometry and quantum science. 80

98 Chapter 6 Dynamical Decoupling Pursuing more sensitive NV magnetometery by probing a large number of NV centers to increase uorescence signal is one of the major focuses of this dissertation. However as discussed in previous chapters, this NV ensemble magnetometry approach has trade-os;for example, in ensemble measurements uorescence from non-resonant NV orientation classes can contribute to the background signal, reducing measurement contrast and subsequently magnetic eld sensitivity. In Chapter 5, we discussed the use of samples with preferentially oriented NV centers in order to recoup this loss in measurement contrast. Another issue comes from the higher NV concentrations typically desired for ensemble measurements compared to those performed on single NV centers. In order to achieve higher NV concentrations, higher concentrations of nitrogen must be incorporated into the diamond sample. The corresponding higher concentration of paramagnetic impurities reduces the NV spin coherence time and subsequently NV magnetic eld sensitivity. Even in samples with low nitrogen concentration, the 81

99 Chapter 6: Dynamical Decoupling 13 C nuclear spins in natural abundance samples limit the NV coherence time to 600 μs [65, 51, 86]. Recently, however, dynamical decouplingtechniques have been employed to reduce the eective interaction of single NV spins with other spin impurities in the environment, specically in impurity environments dominated either entirely by nitrogen electronic spins ( 100 ppm) [90, 64] or entirely by 13 C nuclear spins [91, 71], thereby enabling signicant improvements in the NV single spin coherence lifetime [90, 91, 71] and AC magnetic eld sensitivity [64, 71]. In this chapter, we demonstrate the equally successful application of dynamical decouplingto large numbers of NV spins (> 10 3 ), despite the greater inhomogeneities in local impurity environment and subsequently less optimal average control pulse - delity as compared to addressingsingle NV spins. Employingmultipulse Carr-Purcell- Meiboom-Gill (CPMG) control sequences [92, 93], we realize an order of magnitude extension of the NV ensemble spin coherence time at room temperature in three diamond samples with widely dieringnv densities and spin impurity concentrations, includingtwo impurity environment regimes for which NV dynamical decouplinghas not previously been demonstrated. We also show that the scalingof the NV spin coherence time with the number of pulses in the control sequence depends non-trivially on the concentration of N and 13 C impurities and can dier from the scalingobtained in previous single NV measurements [90, 91, 71]. For some samples, the NV ensemble spin coherence time is increased to > 2 ms, where it begins to be limited by NV spin-lattice relaxation (T 1 6 ms at room temperature). We then apply these dynamical decouplingtechniques to a fourth diamond sample over a range of temperatures (77 K 300 K) in order to suppress both decoherence [3] 82

100 Chapter 6: Dynamical Decoupling and phononic spin relaxation [94]. We demonstrate an extension of the NV spin coherence time to 0.6 s at 77 K, which corresponds to an additional improvement of more than two orders of magnitude compared to previous NV measurements [1, 3] and is on par with the longest coherence times achieved for electronic spins in any solid-state system [95]. Over a wide range of temperatures we also nd that the NV spin coherence time is limited to approximately half of the longitudinal spin relaxation time, a nding that could be relevant to other solid-state spin defects (e.g., P donors in Si). Finally, we demonstrate a ten-fold enhancement in NV ensemble sensing of AC magnetic elds at room temperature, with greater NV magnetic sensitivity enhancement expected at lower temperatures. 6.1 Background As discussed previously, the NV spin environment (i.e., spin bath) is typically dominated by 13 C and N impurities, randomly distributed in the diamond crystal. These spin impurities create time-varying local magnetic elds at each NV spin, which can be approximated as an average eld that uctuates on a time scale set by the mean interaction between spins in the bath, thereby inducing rapid dephasing of freely precessing NV spins on a time scale T 2 1 μs for typical spin impurity concentrations. By applying a single resonant MW π pulse to refocus the dephasing, the spin echo sequence [Figure 6.1(a)] decouples NV spins from bath eld uctuations that are slow compared to the free precession time. Application of additional control pulses, such as n-pulse CPMG and XY sequences, have recently been shown to decouple single NV spins from higher frequency bath uctuations [90, 91, 71]. 83

101 Chapter 6: Dynamical Decoupling a spin echo τ 2 π (π) π ( 2) x x ( 2) x τ 2 b CPMG-n τ 2n τ n τ 2n π (π) π ( 2) y (π) y (π) y (π) y x ( 2) x τ 2n τ n τ 2n c XY-n τ 2n τ n τ 2n π (π) π ( 2) y (π) x (π) x (π) y x ( 2) x τ 2n τ n τ 2n d Uhrig-8 π (π) π ( 2) y (π) y (π) y (π) y (π) y (π) y (π) y (π) y x ( 2) x Figure 6.1: Pulse diagrams of several dynamical decoupling sequences: (a) spin echo, (b) n-pulse CPMG, (c) n-pulse XY, and (d) example Uhrig with n =8pulses. For the dynamical decoupling schemes with periodic pulse spacing, the optimal timing of the control pulses with respect to an AC magnetic eld of frequency f ac = n/(2τ) (green) is shown. CPMG. The CPMG pulse sequence, widely used in the eld of NMR [92, 93], is a simple extension of the spin echo sequence in which the single refocusing π pulse is replaced with n periodically spaced refocusing π pulses. Furthermore, by modulating the phase of the MW carrier signal, the spin rotation axis of the π pulse train is chosen to be perpendicular to that of the rst and last π/2 pulses for the preparation and detection, respectively, of the NV spin superposition state. For example, Figure 6.1(b) shows a CPMG control sequence in which the π/2 pulses are rotated about the x axis and the π pulses are rotated about the y axis. Rotating the NV spin about two dierent axes in this manner allows CPMG pulse sequences containing an even number of pulses to compensate for pulse errors along one spin 84

102 Chapter 6: Dynamical Decoupling rotation axis [the y axis in the Figure 6.1(b) example]. As a result, the CPMG pulse sequence is one of the most robust, given a known initial spin state. However, in many proposed NV applicationsincluding precision magnetometry and quantum information processingthe initial spin state is unknown and two-dimensional pulse error correction is necessary. XY. As shown in Figure 6.1(c), the XY family of pulse sequences extends upon the CPMG sequence by alternating the spin rotation axes of the refocusing π pulses in order to compensate for pulse errors along both the x and y axes, thereby achieving pulse error correction along all directions [96]. As a result, the XY family of pulse sequences is ideal for NV applications such as precision magnetometry and quantum information processing, where the initial spin state is unknown. However, in practical measurements, the CPMG sequence still performs marginally better in situations where the initial spin state is known. Uhrig. It is also worthwhile to mention Uhrig dynamical decoupling sequences [97], which have attracted a fair amount of attention recently [90, 91, 63] as a nonperiodic alternative to the CPMG and XY pulse sequences [Figure 6.1(d)]. However, the Uhrig sequences are most eective at decoupling from noise with a sharp high-frequency cuto [98], making them less suitable for NV centers, which experience a smoother spin bath spectrum. Furthermore, the sequence does not compensate for pulse errors, and the nonperiodic pulse spacing is incompatible with refocusing NV spin dephasing due to 13 C Larmor precession. As a result, this chapter addresses only the periodic CPMG and XY pulse sequences. 85

103 Chapter 6: Dynamical Decoupling 6.2 Dynamical decoupling at room temperature For measurements at room temperature, we used a custom-built wide-eld uorescence microscope (see Appendix B.1 for details) to perform dynamical decoupling on NV ensembles in three diamond samples with dierent NV densities and spin impurity environments. Applying a static eld along one of the diamond crystallographic axes selected approximately one quarter of the NV centers in the 10 μm-diameter detection cross-section to be resonant with the MW pulses. Each diamond sample consisted of an NV-rich layer CVD-grown on a non-uorescing diamond substrate, such that all collected uorescence is attributed to the NV-rich layer Results The NV-rich layer of Sample F was 16 μm thick and contained NV concentration cm 3 ( 60 ppb) [extracted from NV uorescence measurements performed with a confocal microscope (see Appendix B.2 for details)], Nconcentration cm 3 ( 100 ppm) [measured using secondary ion mass spectroscopy (SIMS)], and 1.1% natural abundance 13 C concentration. The high Nconcentration dominated NV decoherence in this sample, limiting the measured spin echo ensemble coherence time to T 2 2 μs. We applied CPMG-n sequences and measured the NV ensemble spin coherence as a function of the full spin evolution time τ; several representative NV ensemble spin coherence decay curves are shown in Figure 6.2. As usual, we extracted T (n) 2 from each decoherence curve by tting the data to a stretched exponential function exp [ (τ/t 2 (n) ) p ] and observed an extension in T (n) 2 by a factor > 10 for n = 128. Furthermore, we found that T (n) 2 exhibited a power-law dependence 86

104 Chapter 6: Dynamical Decoupling 1 Sample F 0.8 Coherence n = 1 n = 2 n = 4 n = 8 n = 16 n = 32 n = Full Spin Evolution Time τ (μs) Figure 6.2: Example NV ensemble coherence decay curves measured using CPMG-n pulse sequences in Sample [ F. ( The solid) lines denote ts to the p ] (n) decoherence envelope of the form exp τ/t 2. on n: T (n) 2 n s, with s =0.65 ± 0.02, which is consistent with the value s 0.67 found recently for single NV centers in similarly nitrogen-rich diamond samples [90]. These results demonstrate that inhomogeneities in the spin bath and MW eld do not limit the eectiveness of dynamical decoupling for extending NV ensemble coherence times by at least an order of magnitude. The NV-rich layers of Samples B and C had much lower NV and N concentrations than found in Sample F: NV cm 3 ( 0.6 ppb), N cm 3 ( 100 ppb) for Sample B and NV cm 2 ( 0.6 ppb), N cm 2 ( 1 ppm) for Sample C. Sample B contained 1.1% natural abundance 13 C such that both N and 13 C contributed to NV decoherence, resulting in periodic collapses and revivals in the NV spin coherence associated with 13 C Larmor precession [41]. In contrast, Sample C was isotopically engineered with 0.01% 13 C such that N impurities dominated NV 87

105 Chapter 6: Dynamical Decoupling n = 1 n = 4 Sample B Coherence n = Full Spin Evolution Time τ (μs) Figure 6.3: Example NV ensemble coherence decay curves measured using CPMG-n pulse sequences in Sample B. Note that the time axis is plotted with a linear scale due to the periodic collapses and revivals in the NV spin coherence associated with 13 C Larmor precession. These collapses and revivals occur in samples where the 13 C abundance is high enough to contribute signicantly to NV decoherence. The solid lines denote ts to the decoherence envelope of the form exp [ ( τ/t 2 (n) ) p ]. 1 Sample C 0.8 Coherence n = 1 n = 4 n = 16 n = 64 n = ,000 10,000 Full Spin Evolution Time τ (μs) Figure 6.4: Example NV ensemble coherence decay curves measured using CPMG-n pulse sequences in Sample [ C. ( The solid ) lines denote ts to the p ] (n) decoherence envelope of the form exp τ/t 2. 88

106 Chapter 6: Dynamical Decoupling decoherence. Despite these dierences in spin impurity environments, we measured similar spin echo NV ensemble spin coherence times (T μs) for the two samples. By applying n-pulse CPMG dynamical decoupling sequences, we extended the NV ensemble spin coherence time [Figures 6.3 and 6.4], achieving T (n) 2 2 ms for both samples, comparable to the longest coherence time reported for dynamical decoupling applied to single NV centers [71]. Discussion Note that the spin impurity environments of Samples B and Cdier signicantly from any diamond samples previously studied with NV dynamical decoupling. For these samples, we observed a power-law dependence of T (n) 2 n s (Figure 6.5), with lower scaling powers than for the nitrogen-rich Sample F: s =0.42 ± 0.02 for Sample B and s =0.35 ± 0.01 for Sample C. These sample-dependent scaling powers and the demonstration of similar behavior for NV single-spin and NV ensemble spin decoherence in samples with comparable impurity environments suggest that multipulse dynamical decoupling control sequences can serve as spectral decomposition probes of spin bath dynamics (e.g., the interactions between the 13 Cand N spin baths [2]). In particular, such studies can provide substantial insight into the eects of environmental inhomogeneity for samples with high NV densities and correspondingly high concentrations of N spin impurities. Such information cannot be extracted from single NV measurements, which require low NV densities to resolve individual NV centers optically. More detail on this spectral decomposition technique as well as some preliminary analysis are presented in Chapter 7. 89

107 Chapter 6: Dynamical Decoupling Coherence Time T 2 (n) (μs) Sample F Sample B Sample C Number of Pulses n Figure 6.5: Scaling of measured NV ensemble coherence time T (n) 2 with number of CPMGpulses n. The solid lines are t functions from which the scaling power were extracted: T (n) 2 n s, where s =0.65 ± 0.02 for Sample F, s =0.42 ± 0.02 for Sample B, and s =0.35 ± 0.01 for Sample C. 6.3 Dynamical decoupling at low temperature At the long 2 ms coherence times achieved using dynamical decoupling techniques at room temperature, the contribution of T 1 processes to decoherence begins to be measurable (see also similar eects in single NV measurements in [91, 71]). In order to extend T (n) 2 further and also to explore the relationship between T 1 and T (n) 2, we apply the same dynamical decoupling techniques to NV ensembles over a range of temperatures below 300 K. For these low temperature measurements, we used a fourth sample, Sample E, which is an isotopically pure (0.01% 13 C) diamond sample with N density cm 3 ( 6 ppb) and NV density cm 3 ( 0.02 ppb). The sample was mounted in a cryostat with no magnetic shielding (see Appendix B.3 for details) 90

108 Chapter 6: Dynamical Decoupling and NV measurements were performed on a detection volume 30 μm 3 ( 100 NV centers) Results Figure 6.6 shows several example NV ensemble spin coherence decay curves measured from Sample Eusing CPMG pulse sequences with dierent numbers of pulses n, at room temperature (300 K) and low temperature (160 K). We extracted T (n) 2 from each decoherence curve and found that for both temperatures, T (n) 2 increases with the number of pulses in the CPMG sequence, limited to approximately half of T 1 :at 300 K, T max 2 =3.3 ± 0.4 ms and T 1 =6.0 ± 0.4 ms; and at 160 K, T max 2 =40± 8 ms and T 1 =77± 5 ms. Figure 6.7 summarizes the dependence of T (n) 2 on the number of CPMG pulses n for several temperatures ranging from liquid nitrogen temperature (77 K) to room temperature (300 K). Notably, at liquid nitrogen temperature, an 8192-pulse CPMG sequence achieved a coherence time of T (n) ms, which is more than two orders of magnitude longer than previous NV coherence time measurements [1, 3]. It also is worthwhile to note that at 160 K, a temperature that can be reached using thermoelectric cooling rather than cryogenic uids, we found T max 2 =40± 8 ms, which is still an order of magnitude longer than any previous measurement of NV spin coherence time. Thus, combined dynamical decoupling and thermoelectric cooling provides a practical way to signicantly increase the NV spin coherence time, which could bene- t many applications of NV ensembles where there is some exibility in the operating temperature, e.g. precision magnetometry [3] and rotation sensing [99, 100]. 91

109 Chapter 6: Dynamical Decoupling a 1 Sample E 300 K 0.8 T 1 Coherence n = 1 n = 4 n = 16 n = 64 n = 256 b T 1 Sample E 160 K Coherence n = 1 n = 4 n = 16 n = 64 n = Full Spin Evolution Time τ (μs) Figure 6.6: Sample E CPMG-n NV spin coherence decay curves (colored data points and solid line ts to data) and longitudinal spin relaxation curves (black data points and dashed line ts to data) at (a) 300 K and (b) 160 K. The T 1 relaxation time measured for each temperature is also indicated in Figure 6.7 (exceptfor 77 K, which had T 1 > 10 s). In the gure insetwe plotthe maximum T (n) 2 achieved, Tmax 2, versus T 1 at each temperature in the interval of K and nd T max 2 =(0.53 ± 0.02)T 1, which diers starkly from the previously expected T (n) 2 limitof 2T 1 [71]. Under the assumption that spin-phonon coupling only causes spin-lattice relaxation (T 1 ), one cannotrecover the measured T max 2 0.5T 1 92

110 Chapter 6: Dynamical Decoupling Coherence Time T 2 (n) (ms) T 2 max (T1 ) = 0.53T T 1 (ms) T 2 max (ms) 300 K 240 K 190 K 160 K 120 K 77 K max T 2 = 580 ms T 1 = 506 ms T 1 = 77 ms T 1 = 30 ms T 1 = 12 ms T 1 = 6 ms Number of Pulses n Figure 6.7: NV ensemble spin coherence time as a function of number of CPMG pulses measured from Sample E over a range of temperatures from K. For temperatures above 77 K, the corresponding measured longitudinal relaxation time T 1 is plotted as a dashed line. The black circle indicates the longest T (n) 2 = 580 ± 210 ms measured for n = 8192 at 77 K. Inset: scaling of the maximum coherence time T max 2 as a function of T 1. limit, even taking into account the possibility of unequal relaxation rates between the three ground-state NV spin sublevels. This can be seen by assuming that only state-changing relaxation processes exist and therefore the decoherence rate between two states a and b, γ2 ab, is given by [101]: ( γ ab 2 = 1 2 k a γ1 a k + γ1 b k k b ), (6.1) where γ j k 1 is the transition rate from state j to state k. Solving rate equations allows one to calculate the measured relaxation rate (γ 1 =1/T 1 ) as a function of these 93

111 Chapter 6: Dynamical Decoupling transition rates, and thus compare it to the decoherence rate (γ 2 =1/T 2 ) given by Equation 6.1. (Note that in this discussion, T 2 refers to the general decoherence time rather than the decoherence time associated with a spin echo measurement as is the convention elsewhere in this dissertation.) Through this approach one can show that for a two-level system, at temperatures ranging from zero to innite temperature, the relation is always T 2 =2T 1. For three or more levels the coecient changes, but the general statement T 2 > T 1 holds (see Appendix C for further details). Therefore we conclude that in our case, since we measure T max 2 < T 1, spin-phonon coupling contributes directly and signicantly to NV spin decoherence [102]. Such phononinduced decoherence is generally relevant to any quantum system in which transitions between two levels can be driven by a two-phonon (Raman) process [94], and could play a role in the coherence properties of many other solid-state defects. At lower temperatures, to be studied in future work, spin-spin interactions should dominate both T 1 and T 2 (n), and thus we expect T max 2 will deviate from the 0.5T 1 limit demonstrated here Discussion T (n) 2 vs n scaling It is evident from Figure 6.7 that the scaling of the coherence time with the number of CPMG pulses varies with temperature. In order to study the scaling behavior of the pure spin environment induced decoherence, we subtracted from the measured decoherence rate the temperature dependent phononic rate 1/(0.53T 1 ) (see inset of Figure 6.7). The corrected coherence time T (n) 2,pure is plotted against the 94

112 Chapter 6: Dynamical Decoupling Pure Coherence Time T 2,pure (n) (ms) K 240 K 190 K 160 K 120 K 77 K Number of Pulses n Figure 6.8: Pure spin environment induced NV decoherence time as a function of the number of pulses in the CPMG pulse sequence. The data are t to a power-law function (black line) from which a scaling behavior of T (n) 2,pure n (0.67±0.03) is extracted. number of CPMG pulses n in Figure 6.8, exhibiting striking universal behavior for all temperatures. We t the corrected data to a power law scaling, and nd T (n) 2,pure n (0.67±0.03). This value is consistent with the expected scaling power of 2/3 for a Lorentzian noise spectrum of an electronic spin bath [2, 103]. It is worthwhile to note that the same analysis may be performed on the previous room temperature CPMG measurements of Samples B and C, which were likely limited by T 1. Extracting the pure coherence time T (n) 2,pure yields corrected scaling powers of s =0.59 ± 0.02 for Sample B and s =0.53 ± 0.02 for Sample C, both of which are still less than the s = 2/3 limit for a Lorentzian noise spectrum. This behavior is addressed in more detail in Chapter 7. 95

113 Chapter 6: Dynamical Decoupling Measurements on other samples Even though our measurements were performed on an isotopically pure sample (0.01% 13 C), we expect that similar results can be obtained for natural abundance diamond (1.1% 13 C), since the dynamical decoupling sequences we employ are also eective in suppressing dephasing caused by the nuclear spin bath [91, 3]. However, as discussed in Chapter 3, special care must be taken in aligning the applied static magnetic eld along the NV axis for natural abundance diamond to avoid additional NV spin decoherence caused by variations in the eective Larmor frequency of nearby 13 C nuclear spins due to magnetic eld misalignment [86, 77]. At present we have performed measurements on one additional sample, the isotopically pure (0.01% 13 C) Sample C. Due to the higher N density of the sample, the coherence times achieved were consistently lower. At 77 K with n = 4096 pulses, we obtained a coherence time of T (n) 2 =50± 9 ms, with a scaling of the coherence time with the number of pulses of T (n) 2 n (0.54±0.01), which is consistent with the s =0.53 ± 0.02 corrected scaling power determined by removing T 1 relaxation eects from the room temperature CPMG NV coherence measurements. From this reproducibility and the ensemble nature of our experiments, we conclude that the data presents the behavior of typical NV centers, averaged over inhomogeneous broadening eects such as strain. Pulse errors The CPMG pulse sequence is quite robust against noise (e.g. external magnetic eld uctuations) and pulse errors, allowing us to apply thousands of pulses. How- 96

114 Chapter 6: Dynamical Decoupling 2 Normalized Contrast K 240 K 190 K 160 K 120 K 77 K Number of Pulses n Figure 6.9: Measured normalized signal contrast as a function of the number of CPMG pulses n for dierent temperatures. ever, we noticed a modest reduction in signal contrast and increased noise for pulse sequences of 10, 000 pulses or more (Figure 6.9). The resulting reduction in signalto-noise, together with the long integration times needed, currently prevents us from increasing the coherence time to beyond 0.6 s. In future work we will address these issues using various approaches, including more robust composite pulses [104] that could be better suited to manipulate an inhomogeneously broadened ensemble. Outlook Achieving long NV spin coherence times in diamond samples with high impurity concentration ( 1 ppm) is a crucial step toward creating nonclassical states of NV ensembles. Such non-classical states could form the basis for high-sensitivity quantum metrology, potentially allowing signicantly improved sensitivity and bandwidth [105, 97

115 Chapter 6: Dynamical Decoupling 106], and could also serve as a resource for quantum information protocols. To observe signicant entanglement between neighboring NV centers, their decoherence rate must be small compared to the frequencyassociated with their interaction. For a realistic diamond sample with N 1 ppm and NV 10 ppb, coherence times larger than 50 ms are needed for signicant entanglement, which is within reach given the results presented here. For example, collective spin squeezing using L 2 z one-axis squeezing techniques [107] could be created through application of pulse sequences that average out the x, y components of the spin-spin dipolar coupling [108]. Such pulse sequences can be straightforwardlyapplied in conjunction with the CPMG pulse sequences used here for extending the NV spin coherence time. 6.4 Multipulse magnetometry Now we take advantage of the extension in NV ensemble spin coherence time achieved bymultipulse dynamical decoupling sequences to demonstrate up to an order of magnitude enhancement in AC magnetic eld sensitivity. These preliminary measurements were performed at room temperature and were subsequentlylimited bythe millisecond time scale of NV spin relaxation; however, the multipulse magnetometrytechnique can be easilyperformed at lower temperatures to further improve magnetic sensitivity Sensitivity analysis Recall, in a standard AC magnetometrymeasurement utilizing a spin echo sequence, an oscillating magnetic eld, b(t) = B sin [(2π/τ ac ) t + ϕ 0 ], induces a net 98

116 Chapter 6: Dynamical Decoupling phase accumulation of the NV spin coherence, which is maximized when the full time of the spin echo sequence is equivalent to the period of the AC magnetic eld (τ ac ) and the phase oset ϕ 0 is such that the control pulses coincide with nodes in the magnetic eld [Figure 6.1(a)]. Under these conditions, the eld amplitude B can be extracted from the measurement of accumulated NV spin phase with optimum sensitivity normalized per unit time, given approximately by [52]: η se π 1 exp 2gμ B α 0 βτac [( τac T 2 ) p ]. (6.2) As before, the number of photons collected per measurement β is dependent on the optical collection eciency and number of NV spins contributing to the measurement. The initial contrast α 0 is modied by NV decoherence T 2 over the course of the spin echo measurement, described phenomenologically by an exponential factor with power p. The value of p is found to be sample dependent, in the range of 1 to 3, and is related to the dynamics of the spin environment and to ensemble inhomogeneous broadening [2] (see Chapter 7). In AC magnetometry measurements which employ n-pulse dynamical decoupling sequences, the sensitivity is given approximately by Equation 6.2 with two modications: the measurement time is increased (τ ac n 2 τ ac) and the NV ensemble spin coherence time is extended (T 2 T (n) 2 =T 2 n s ). The resulting sensitivity is given by: η (n) π 1 2gμ B α 0 β n τ exp 2 ac [( n (1 s) τ ac 2T 2 ) p ]. (6.3) Because the measurement time increases linearly with the number of control pulses n, whereas the coherence time increases sub-linearly, there is an optimum number of pulses n opt that yields the most sensitive measurement of an AC magnetic eld of 99

117 Chapter 6: Dynamical Decoupling period τ ac given a set of sample-determined parameters: [ ( ) p ] 1 1 2T2 p(1 s) n opt =. (6.4) 2p(1 s) For a given sample, all the parameters except τ ac are xed and we can simplify Equation 6.4 to n opt (1/τ ac ) 1 (1 s). From this relationship, we see that at higher AC frequencies, more pulses are needed to reach the optimum sensitivity, which can be understood intuitively by realizing that the high frequency regime corresponds to short time intervals between control pulses during which time there is very little contrast lost due to decoherence (τ ac T 2 ). More pulses increase the sensitivity by allowing for a longer measurement time and subsequently more phase accumulation per measurement. This intuition also illustrates why multipulse sequences are more eective at enhancing magnetometry sensitivity in the high frequency regime, where the spin echo scheme provides relatively poor magnetic eld sensitivity (Figure 6.12). As discussed in the previous section, extension of the NV ensemble spin coherence time via multipulse dynamical decoupling (and thus enhancement of magnetic eld sensitivity) is eventually limited by NV spin-lattice relaxation (T 1 ), beyond which τ ac increasing the number of control pulses is ineective. In applications where it is compatible, however, a modest amount of cooling can be applied to further improve magnetic eld sensitivity Results We used Sample C mounted in the wide-eld uorescence microscope setup to perform NV ensemble magnetometry measurements comparing spin echo and multipulse dynamical decoupling schemes. The detection volume was approximately 30 μm 3, 100

118 Chapter 6: Dynamical Decoupling spin echo δb Fluorescence Contrast S n = 28 n = 54 δs δb δb spin echo Field Amplitude B (nt) Figure 6.10: Examples of measured normalized uorescence signals as functions of AC eld magnitude B using a spin echo sequence (green) (wider amplitude range shown in the inset) and multipulse XY sequences with 28 (red) and 54 (blue) control pulses, illustrating how the uncertainty in the measured signal (δs) limits the uncertainty in the extracted magnetic eld magnitude (δb). The sine behavior of the signal with respect to b ac is achieved by shifting the phase of the last microwave pulse by 90 from what is shown in Figure 6.1(c). corresponding to 10 3 sensing NV centers aligned along the static magnetic eld. We employed n-pulse XY sequences to provide more isotropic compensation for pulse errors [96]. Note that though Sample C is an isotopically engineered sample with 0.01% 13 C, such pulse sequences are equally applicable for samples with a natural abundance of 13 C with the additional complication that the period of the AC eld to be measured must correspond to a 13 C revival in the NV spin decoherence curve. Figure 6.10 shows magnetometry curves of the measured NV uorescence as a function of applied AC magnetic eld amplitude for spin echo, 28-pulse, and 54-pulse sequences in order to illustrate how the magnetic eld sensitivity improves with mul- 101

119 Chapter 6: Dynamical Decoupling 18 f ac = 84.5 khz 16 Sensitivity η (n) (nt/ Hz) Number of Pulses n Figure 6.11: Measured multipulse magnetic eld sensitivity as a function of number of pulses for an 84.5 khzac eld (circles). The solid line is calculated using Equation 6.3 with experimental parameters α 0 = 0.055, β =3.4 photons per measurement, T 2 = 250 μs, p =1, and s =0.37. tipulse sequences compared to a spin echo sequence. In essence, the larger NV spin phase accumulation achieved per measurement in the multipulse scheme increases the maximum slope S B of the magnetometry curve such that the same uctuation in signal δs corresponds to a smaller uncertainty in the magnetic eld δb. The AC magnetic eld sensitivity improved with the number of pulses n, in good agreement with predicted values up to n 150 pulses, at which point pulse errors began to degrade the measured sensitivity (Figure 6.11). While for the present measurements the AC magnetic eld consisted of a locked, single frequency component, the procedure can be extended to nite-bandwidth elds [64]. Over a wide range of AC magnetic eld frequencies our NV ensemble magnetometry measurements conrmed that multipulse dynamical decoupling outperformed the spin echo scheme, in agreement with theoretical expectations (Figure 6.12). The en- 102

120 Chapter 6: Dynamical Decoupling 10 7 η se η (n) Sensitivity η (n) (nt/ Hz) Field Frequency f ac (Hz) Figure 6.12: Comparison of calculated (lines) and measured (points) magnetic eld sensitivity at several AC eld frequencies,for spin echo and multipulse XY sequences (using the measured optimum number of pulses for each frequency). hancement in magnetic eld sensitivity provided by multipulse dynamical decoupling was especially pronounced at frequencies higher than the spin echo 1/T 2 coherence. For example,at a frequency of 220 khz,we measured a factor of 10 improvement in magnetic eld sensitivity: from η se =67.7 ± 3.5nT/ Hz using a spin echo sequence, to η (n) =6.8 ± 0.4nT/ Hz using a 240-pulse XY sequence. 6.5 Summary In summary,we experimentally demonstrated that multipulse dynamical decoupling control sequences can signicantly improve NV ensemble spin coherence times and enhance NV magnetic sensitivity to AC elds. At room temperature,we observed order of magnitude coherence time extension in samples with widely diering 103

121 Chapter 6: Dynamical Decoupling NV densities and spin impurity environments. For these spin environments that dier from those studied previously, we found signicantly dierent scaling of the NV ensemble spin coherence time with the number of pulses in the control sequence. These results provide insight into spin bath dynamics and could guide future applications in multi-spin quantum information processing and metrology. By measuring at lower temperatures, we demonstrated more than two orders of magnitude further improvement in the NV coherence time: up to T sby combining dynamical decoupling control sequences with cryogenic cooling to 77 K; and T 2 40 ms for temperatures achievable via thermoelectric cooling (> 160 K). By studying the dependence of coherence time T (n) 2 and spin relaxation time T 1 on temperature, we identied an eective limit of T max 2 0.5T 1, which we attribute to phonon-induced decoherence. Given this limit, we expect that for low NV densities T 2 on the order of a few seconds should be achievable at liquid nitrogen temperatures. We also showed that multipulse dynamical decoupling improves NV ensemble AC magnetic eld sensitivity relative to the spin echo scheme, with a ten-fold enhancement for higher frequency elds at room temperatures, with greater enhancement expected at lower temperatures. Using samples with higher NV densities could further improve this sensitivity; however, at high NV concentrations ( 1 ppb), the coherence time may be limited by NVNV interactions, since CPMG pulse sequences aect both the NV spin and its NV spin bath at the same time, which cancels the decoupling eect. Other techniques, such as WAHUHA and MREV pulse sequences [109, 110], may be applied to address this issue. The greatly extended NV spin coherence time presented in this work, which does 104

122 Chapter 6: Dynamical Decoupling not require an optimally chosen NV center, could also form the building block for wide-ranging applications in quantum information, sensing, and metrology in the solid-state [111, 65].In particular, the fact that such long coherence times can be achieved with high-density NV ensembles suggests that spin squeezing and highly entangled states can be created, since T 2 > NVNV dipolar interaction time.finally, this work could provide a key step toward realizing interaction-dominated topological quantum phases in the solid-state, as well as a large family of driven many-body quantum Hamiltonians [112]. 105

123 Chapter 7 Spectral Decomposition In the previous chapter, we applied dynamical decoupling techniques to NV ensembles in several diamond samples with dierent spin impurity environments. In each sample, the multipulse control sequences were able to signicantly extend the NV ensemble spin coherence time; however, we noted that the scaling ofthe coherence time as a function ofthe number ofpulses varied from sample to sample. These observed variations suggest that the application ofthese pulse sequences to extract information about the spectral function of the bath may reveal sample-dependent spin bath dynamics. In this chapter, we employ these pulse sequences as a general spectral decomposition technique [113], which has previously been demonstrated in other spin systems [114, 115], to characterize the dynamics ofthe composite spin bath which makes up the NV spin environment, consisting ofboth electronic spin (N) and nuclear spin ( 13 C) impurities. We study three diamond samples with a range ofnv densities and impurity spin concentrations (measuring both NV ensembles and single NV centers) 106

124 Chapter 7: Spectral Decomposition and nd unexpectedly slow dynamics for the electronic spin baths in two diamond samples with natural abundance (1.1%) of 13 C nuclear spin impurities. We identify a possible new mechanism in diamond involving an interaction between the electronic and nuclear spin baths that can explain the observed suppression of electronic spin bath dynamics. Gaining a better understanding of the sources of NV spin decoherence and their interplay by using techniques such as spectral decomposition will inform the optimization of future samples for magnetic sensing. 7.1 Background Spectral decomposition technique Due to coupling of spins (here, NV electronic spins) to their magnetic environment, coherence is lost over time with the general form C(t) =e χ(t), where the functional χ(t) describes the time dependence of the decoherence process. In the presence of a modulation acting on the NV spins (e.g., a resonant MW pulse sequence), described by a lter function in the frequency domain F t (ω), the decoherence functional is given by [116, 117]: χ(t) = 1 π 0 dωs(ω) F t(ω) ω 2, (7.1) where S(ω) is the spectral function describing the coupling of the system to the environment. Equation 7.1 holds in the approximation of weak coupling of the NV spins to the environment, which is appropriate for systems with (dominantly) electronic spin baths [118]. Thus, we restrict our measurements in this work to samples that meet this criterion of nitrogen-dominated spin baths. 107

125 Chapter 7: Spectral Decomposition S(ω) can be determined from straightforward decoherence measurements of the NV spin using a spectral decomposition technique [115, 114, 113]. As seen from Equation 7.1, if an appropriate modulation is applied to the NV spins such that F t (ω) ω 2 t = δ(ω ω 0 ) (7.2) then χ(t) = 1 π ts(ω 0). (7.3) Therefore, by measuring the time dependence of the spin coherence C(t) when subjected to such a spectral δ-function modulation, we can extract the spin bath's spectral component at frequency ω 0 : S(ω 0 )= π ln(c(t))/t. (7.4) This procedure can then be repeated for dierent values of ω 0 to provide complete spectral decomposition of the spin environment. A close approximation to the ideal spectral lter function F t (ω) described above can be provided by the CPMG pulse sequence for dynamical decoupling of a spin from its environment [93]. Recall, the CPMG pulse sequence is an extension of the spin echo sequence [119], with n equally spaced π pulses applied to the system after initially rotating it into the x axis with a π/2-pulse. We apply a deconvolution procedure to correct for deviations of this lter function from the ideal Dirac δ-function Spectral decomposition deconvolution procedure The coherence of a two-level quantum system can be related to the magnitude of the o-diagonal elements of the system's density matrix. Specically, we deal here 108

126 Chapter 7: Spectral Decomposition CPMG Filter Function F n (ω) (arb) pulse 64 pulses 128 pulses Frequency ω (10 6 rad/s) Figure 7.1: CPMG lter functions. Calculated lter function Fn CPMG (ω) for three examples of CPMG-n pulse sequences (for n =1, 64, 128). with NV electronic spins in a nite external magnetic eld, which can be treated as eective two-level spin systems with quantization (z)axis aligned with the NV axis. When the NV spins are placed into a coherent superposition of spin eigenstates (e.g., aligned with the x axis of the Bloch sphere)the measurable spin coherence is given by C(t) =Tr[ρ(t)S x ]. The lter function for the n-pulse CPMG control sequence Fn CPMG (ω) covers a narrow frequency region (with a width given by π/t, where t is total length of the sequence)which is centered at ω 0 = πn/t, and is given by [120]: ( ) ( ) ωt sin 4 ωt Fn CPMG (ωt) =8sin 2 4n 2 cos ( ). (7.5) 2 ωt 2n We note that the narrow-band feature of the CPMG lter essentially denes the eectiveness of this sequence for dynamical decoupling. The lter function Fn CPMG (ωt) associated with the CPMG-n pulse sequence deviates from a δ-function by the nite width of the main spectral peak and by the presence of higher harmonics (see Figure 7.1). In addition, the central frequency of the lter function changes with the duration of the experiment and with the number 109

127 Chapter 7: Spectral Decomposition of pulses. Therefore, reconstructing the spin bath spectral function S(ω) from the decoherence data requires a solution of a Fredholm type equation to extract S(ω) (Equation 7.1): χ(t) = 1 π 0 dωs(ω) F t(ω) ω 2. (7.6) This extraction is accomplished by assuming that the spectral function decays to zero at high frequency and by noting that the lter function includes high harmonics but negligible low frequency artifacts. Thus, the procedure starts at the high frequency values of χ(ω), and works back to lower frequencies by subtracting the eect of higher harmonics using the reconstructed high frequency values of S(ω) and the analytical form of the lter function. Due to the high frequency components of the lter function and the monotonically decreasing nature of the spectral function, a naive reconstruction of the spectral function assuming a δ-function form of the lter function produces results that are biased to lower values by 15%. This bias is removed using the reconstruction algorithm presented above Spectral function of a spin bath The composite spin environment in diamond is dominated by a bath of uctuating N electronic spin impurities, which causes decoherence of the probed NV electronic spins through magnetic dipolar interactions. In the regime of low external magnetic elds and temperatures well above the Zeeman energy of the N electronic spins (relevant to the present experiments), the N bath spins are randomly oriented, and their ip-ops (spin state exchanges) can be considered as random uncorrelated events 110

128 Chapter 7: Spectral Decomposition [118]. Therefore, the resulting spectrum of the N bath's coupling to the NV spins can be assumed to be Lorentzian [116]: S(ω) = Δ2 τ c π 1 1+(ωτ c ) 2. (7.7) This spin bath spectrum is characterized by two parameters: Δ is the average coupling strength of the N bath to the probed NV spins, and τ c is the correlation time of the N bath spins with each other, which is related to their characteristic ip-op time. In general, the coupling strength Δ is given by the average dipolar interaction energy between the bath spins and the NV spins, and the correlation time τ c is given by the inverse of the dipolar interaction energy between neighboring bath-spins. Since such spin-spin interactions scale as 1/r 3, where r is the distance between spins, it is expected that the coupling strength scales as the N bath spin density n N (i.e., Δ n N ), and the correlation time scales as the inverse of this density (i.e., τ c 1/n N ). Also recall from Chapter 6 that the multipulse CPMG sequence used in the spectral decomposition technique extends the NV spin coherence lifetime by suppressing the time-averaged coupling to the uctuating spin environment. In general, the coherence lifetime T 2 (n) increases with the number of pulses n employed in the CPMG sequence. For a Lorentzian bath, in the limit of very short correlation times (τ c T 2 ), the sequence is inecient and T 2 (n) n 0 (no improvement with number of pulses). In the opposite limit of very long correlation times (τ c T 2 ), the scaling is T 2 (n) n 2/3 [53, 103, 64] (see also recent work on quantum dots [121]). In the following we apply spectral decomposition to study the spin bath dynamics and resulting scaling of T 2 (n) with n for NV centers in diamond. 111

129 Chapter 7: Spectral Decomposition 7.2 Results We applied the spectral decomposition technique to extract the spin bath parameters Δ and τ c for three diamond samples with diering concentrations of electronic and nuclear spin impurities. We rst study Sample C, which has a very low concentration of 13 C nuclear spin impurities (0.01%), a moderate concentration of N electronic spin impurities ( 1 ppm), and a moderate NV density ( 0.6 ppb) with spin echo coherence time T μs. Recall from Chapter 6 that even for few numbers of pulses, phononic relaxation likely contributes strongly to the NV spin decoherence of this sample at room temperature. As a result, we perform measurements on Sample C at liquid nitrogen temperature in a cryostat setup (see Appendix B.3 for details). We then employed a wide-eld CCD setup (see Appendix B.1 for details) to measure Sample F, a CVD diamond sample with a 10μm thick N-doped layer containing natural 13 C concentration (1.1%), high N concentration ( 100 ppm), and large NV density ( 0.1 ppm). Finally, we study Sample G, a type 1b HPHT diamond containing natural 13 C concentration, high N concentration ( 50 ppm), and low NV density ( 0.01 ppb). Due to its low NV density, we studied Sample G using a confocal microscope able to measure single NV centers (see Appendix B.2 for details). Note that measurements on Samples F and G were performed at room temperature because both samples had short NV spin coherence times which were not limited by longitudinal spin relaxation even with large numbers of pulses. We use two methods of analysis to characterize the underlying spin bath spectrum S(ω) in terms of the coupling strength Δ of the bath spins to the probed NV spins and the correlation time τ c of the bath spins with each other: in the rst we separately 112

130 Chapter 7: Spectral Decomposition Coherence n = 1 n = 4 n = 16 n = 64 n = 256 n = Full Spin Interrogation Time t (ms) Figure 7.2: Examples of NV ensemble spin coherence as a function of time C n (t) measured at 77 K for CPMGpulse sequences with dierent numbers of pulses n. t each spectral function S n (ω) to a Lorentzian, which provides individual-t values for Δ and τ c for each of the CPMG-n pulse sequences, and in the second we average all the S n (ω) values for each ω and then t the resulting mean spectral function S n (ω)) n to a Lorentzian, which provides single, average-t values for Δ and τ c Sample C We rst discuss the results for 12 C isotopically enriched sample. Figure 7.2 plots examples of the measured NV ensemble spin coherence decay C n (t) as a function of pulse sequence duration t for CPMGpulse sequences with dierent numbers of π pulses n. The measured C n (t) are well described by a stretched exponential e (t/t 2 (n) ) p, where the value of p varies between 1 and 3 related to dynamics of the spin environment and ensemble inhomogeneous broadening. This behavior is consistent with a Lorentzian spin bath spectrum and indicates that N spin impurities are the dominant 113

131 Chapter 7: Spectral Decomposition Bath Spectrum Sn(ω) (10 3 arb) Δ = 8-30 khz τ c = μs Δ = (12 ± 1) khz τ c = (52 ± 11) μs Frequency ω (10 6 rad/s) Figure 7.3: Application of the spectral decomposition technique to Sample F. Derived values for the spin bath spectral functions S n (ω) for all CPMG pulse sequences (blue dots) and binned average values (red crosses). Each blue dot represents a pulse sequence with a specic duration t and number of pulses n, such that the probed frequency is ω = πn/t. Also shown is the best t Lorentzian for the mean spectral function S n (ω) n (solid black line), and a range of best-t Lorentzians for the individual spectral functions S n (ω) for each CPMGpulse sequence (light blue band). source of NV spin decoherence. Note that the second most important electronic spin impurities are NVH defects, which are an order of magnitude less abundant than N impurities [8]. The contribution of instantaneous spin diusion due to interactions between neighboring NV spins and MW-induced spin ips [122] is also negligible, as the NV Rabi frequency for typical MW power ( 10 MHz) is large compared to the NV electronic spin transition linewidth ( 3 MHz), and the NV dipolar coupling time scale (proportional to the inverse of the dipolar coupling energy) is at least an order of magnitude longer than the longest measured NV spin coherence time. Using the measured C n (t) data we derive values for the spin bath spectral function S n (ω) for each CPMGpulse sequence by deconvolving χ n (ω) with the lter function 114

132 Chapter 7: Spectral Decomposition 10 2 Coherence Time T 2 (n) (ms) data fit to data simulated Number of Pulses n Figure 7.4: Scaling of T 2 (n) with the number n of CPMG pulses: derived from NVspin coherence decay data C n (t) (dots); t of dots to a power law T 2 (n) =(0.54 ± 0.02 ms)n (0.54±0.01) (solid line); and synthesized from the average-t Lorentzian spin bath spectrum with Δ=12kHz, τ c =52μs (open squares), which yields a consistent t T 2 (n) =(0.54 ± 0.08 ms)n (0.59±0.03). Fn CPMG (ωt). The resulting values for the spectral functions S n (ω) for all CPMG pulse sequences (i.e., all values of n) are plotted together in Figure 7.3. Note that the average data (red crosses) in Figure 7.3 are calculated by binning the full data set for S n (ω) into 100 frequency values between 0 and 1 MHz. We then extracted the best-t Lorentzian spin bath spectrum (t to the average of all data points), yielding a coupling strength of Δ = 12 ± 1 khz and correlation time τ c =52± 11 μs, which agrees reasonably with the range of values we nd for the Lorentzian spin bath spectra S n (ω) t to each CPMG pulse sequence individually, Δ 8 30 khz and τ c μs. These values are in reasonable agreement with the expected N dominated bath values for Δ and τ c for this sample's estimated concentrations of 13 C and N spins (see Table 7.1), indicating that N electronic spin impurities are the dominant source of NVdecoherence. In Figure 7.4 we plot the 115

133 Chapter 7: Spectral Decomposition Table 7.1: Comparison of key characteristics and extracted average-t Lorentzian spin bath parameters for the samples studied in this work. Sample C Sample F Sample G Meas. technique ensemble (77K) ensemble single NV N concentration 1 ppm 100 ppm 50 ppm NV density 0.6 ppb 0.1 ppm 0.01 ppb 13 C concentration 0.01% 1.1% 1.1% Spin echo T 2 (520 ± 20) μs (2 ± 1) μs (5 ± 1) μs T (n) 2 scaling n (0.54±0.01) n (0.65±0.05) n (0.7±0.1) Δ (expected) 60 khz 6 MHz 3 MHz Δ (measured) (12 ± 1) khz (7 ± 3) MHz (1 ± 1) MHz τ c (expected) 15 μs 0.17 μs 0.34 μs τ c (measured) (52 ± 11) μs (3 ± 2) μs (10 ± 5) μs NV ensemble spin coherence lifetime T (n) 2 as a function of the number n of CPMG π pulses. We found that the scaling behavior extracted from measurements of the coherence decay C n (t) and synthesized from the average-t Lorentzian spin bath spectrum with Δ=12kHz, τ c =52μs are in reasonable agreement Samples F and G We repeated the spectral decomposition analysis for the two other diamond samples, with the results for all three samples summarized in Figure 7.5 and Table 7.1. We nd reasonable agreement between the measured and expected values for the 116

134 Chapter 7: Spectral Decomposition a Bath Spectrum Sn(ω) (arb) Frequency ω (rad/s) b Coherence Time T 2 (n) (ms) Number of Pulses n Sample C Sample F Sample G Figure 7.5: Comparison of spectral decomposition results from Samples C (blue), F (green) and G (red). (a) Spin bath spectral functions data for all CPMG pulse sequences (dots), and associated Lorentzian ts using the average-t method (solid lines) and the individual-t method (color bands). (b) Scaling of T (n) 2 with the number of CPMG pulses n as extracted from NV spin coherence decay data C n (t) (dots) and as synthesized from the average-t Lorentzian spin bath spectrum (open squares). The solid lines are power-law ts to the data with scaling powers: T (n) 2 n (0.54±0.01) for Sample C, T (n) 2 n (0.65±0.05) for Sample F, T (n) 2 n ( ) for Sample G. 117

135 Chapter 7: Spectral Decomposition NVspin bath coupling strength Δ in all three diamond samples, with Δ scaling approximately linearly with the N concentration. As mentioned before, the measured and expected values for the spin bath correlation time τ c agree reasonably well (within a factor of 3) for Sample C. However, we nd a striking discrepancy between the measured and expected values of τ c for the two samples with 1.1% 13 C concentration (Samples F and G): both samples have measured spin bath correlation times that are more than an order of magnitude longer than that given by the simple electronic spin bath model, though the relative values of τ c for these two samples scale inversely with N concentration, as expected. 7.3 Discussion Spin bath suppression We explain the observed suppression of spin bath dynamics as a result of random, relative detuning of electronic spin energy levels due to interactions between proximal electronic (N) and nuclear ( 13 C) spin impurities (similar to processes identied by Bloembergen [123] and Portis [124] for other solid-state spin systems). The ensemble average eect of such random electronic-nuclear spin interactions is to induce an inhomogeneous broadening ΔE of the resonant electronic spin transitions in the bath, which reduces the electronic spin ip-op rate R ( 1/τ c ) given by [125, 126] R π 9 Δ 2 N ΔE, (7.8) where Δ N is the dipolar interaction between N electronic spins. In this physical picture, ΔE is proportional to the concentration of 13 C impurities and to the N 13 C 118

136 Chapter 7: Spectral Decomposition 250 Relative N Spin Frequency Shift (MHz) Pair Index Number Figure 7.6: Monte Carlo simulation of the relative frequency shift between 1000 pairs of N electronic spins induced by hyperne interactions with 13 C nuclear spin impurities that are randomly distributed in a diamond lattice at the natural abundance concentration of 1.1%. hyperne interaction energy, whereas Δ N is proportional to the N concentration. In Samples F andg, the N 13 C hyperne interaction is dominated by the strong contact term (comparedto the weaker dipolar contribution, see [127, 128]). Therefore, the inhomogeneous broadening energy ΔE does not scale simply with the concentration of 13 C. To make a realistic estimate of ΔE, we carriedout a Monte-Carlo simulation of 1000 pairs of N spin impurities, each with a dierent random distribution of 13 C spin impurities in a diamond lattice at their natural abundance of 1.1%. The calculatedrelative electronic spin resonance frequency shift for each N pair is plottedin Figure 7.6. We ndthe average value of this detuning 9.5 MHz andthe standard deviation 30 MHz, with a few rare events in which a 13 C occupies the nearest lattice to the N, producing a frequency shift 300 MHz. Thus, we estimate ΔE 10 MHz and Δ N 1 MHz for Samples F andg. These values imply an order of magnitude suppression of R comparedto the bare electronic spin ip-op rate 119

137 Chapter 7: Spectral Decomposition CVD Layer NV Layer HPHT Layer Counts (arb) NV Fluorescence Reflected Light Depth (μm) Figure 7.7: Confocal scan of NV uorescence and hence concentration (red) as a function of depth in Sample C. The green line indicates the surfaces of the sample as measured by enhanced reection of the green excitation light. ignoring N 13 C interactions (R bare Δ N ), which is consistent with our experimental results for τ c 1/R (see Table 7.1). For Sample C, the low concentration of 13 C spins greatly reduces the magnitude of ΔE, and hence the spin bath suppression is negligible Ensemble averaging Independent measurements of N and NV concentration were performed on each sample to conrm that sample inhomogeneities are within acceptable ranges and therefore do not greatly aect the results of the spectral decomposition analysis. Sample C has a low N concentration that is below the detection threshold of the standard secondary ion mass spectroscopy (SIMS) technique; as a result, we performed confocal scans on the sample to measure the spatial distribution of NV centers (see Figure 7.7). The sample consists of a type 1b HPHT substrate with typical high N 120

138 Chapter 7: Spectral Decomposition and low NV concentration (corresponding to the μm range in Figure 7.7) which was overgrown with a CVD layer 160 μm thick. The majority ofthe NV uorescence comes from a 60 μm thick region within the CVD layer (corresponding to the μm range in Figure 7.7). Consequently, the measurements performed on this sample exclusively probe this region, which shows less than a factor of two variation in NV concentration throughout its depth. With the reasonable assumption that N and NV concentrations are proportional to each other, we conclude that the N concentration probed in this sample is similarly homogeneous to within about a factor of two. We performed SIMS measurements on Sample F to characterize inhomogeneity in the N concentration (see Figure 7.8). Again assuming the N and NV concentrations are locally proportional to each other, we determined the mean N concentration probed by NV uorescence measurements, n N, from a weighted average over the local N concentration n N (z): Z 0 n N = dzn2 N (z) Z dzn 0 N(z). (7.9) Using the SIMS data we nd n N (2.6±1.7) cm 3 ( 150±100 ppm), which is consistent with the coupling strength Δ extracted using the spectral decomposition technique and with Monte Carlo simulations ofthe spin environment. In addition, our measurements were carried out on a conned lateral region 10 μm 10 μm to suppress the eects ofinhomogeneities on a larger scale. The SIMS measurements also indicate that within the detection limits (1 ppm) there are no other extrinsic defects present, such as Boron or Si. As is typical oftype 1b HPHT diamonds, Sample G contains a high concentra- 121

139 Chapter 7: Spectral Decomposition 6 N Concentration (10 19 cm -3 ) Depth (μm) Figure 7.8: Inhomogeneity of N concentration in Sample F as measured using secondary ion mass spectroscopy. tion of N impurities ( 50 ppm) and very lowconcentration of NV centers, allowing for single NV measurements using a confocal apparatus. As a result, possible inhomogeneities of the N concentration do not aect the measured data. The sample was characterized by SIMS to have an N concentration of approximately ppm, consistent with the value for coupling strength Δ extracted using the spectral decomposition technique. For all three samples, the spectral decomposition technique yielded values for coupling strength Δ which are consistent with independent measurements and estimates of the N concentration, suggesting that sample inhomogeneities are within expected ranges and understood to within a factor of 2-3. As a result, we have condence that the values for the N spin bath correlation time τ c provided by the spectral decomposition technique are, for Samples F and G, an order of magnitude longer than the expected values given by a simple model of the N spin bath that ignores N 13 C interactions. Furthermore, the results we obtained via spectral decomposition could not 122

140 Chapter 7: Spectral Decomposition be recreated with any combination of Δ and τ c that follow the expected scaling with N concentration, regardless of the actual value of the assumed concentration and even when inhomogeneous eects are taken into account. Therefore, the combined results from the three samples, along with the independent measurements of N concentration and the consistency of the extracted values for Δ, strongly support the validity of the unexpectedly long correlation time τ c observed in Samples F and G. 7.4 Summary In summary, we applied the spectral decomposition technique to three diamond samples with dierent composite spin environments in order to characterize their spin bath dynamics. For samples with a nite concentration of 13 C nuclear spin impurities, this technique revealed an order of magnitude suppression of the N electronic spin bath dynamics, which can be explained by random interactions between proximal electronic and 13 C nuclear spin impurities. This spin bath suppression enhances the ecacy of dynamical decoupling in samples with high N impurity concentration and therefore has important implications in achieving optimized NV ensemble magnetometry. Recall that NV magnetic sensitivity is dependent on the product of NV spin coherence time and the number of photons collected per measurement, where the photons collected is typically proportional to the number of NV centers probed, which is in turn proportional to the N concentration. In this work, we realized a product of NV spin density and coherence time of ms/cm 3 in Sample F at room temperature due to spin bath suppression. This spin density-coherence time product is within an order of magnitude of the state-of-the-art product achieved in atomic 123

141 Chapter 7: Spectral Decomposition systems [129]. Further optimization may be possible by engineering the spin bath suppression, e.g., with 13 C concentration higher than the natural value. Such optimization, along with spectral decomposition studies of samples with varying 13 C and N concentrations, in a range of temperatures, andat high magnetic elds, will be pursuedin future work. Finally, it is also worthwhile to note that the spectral decomposition technique presentedhere, basedon well-known pulse sequences anda simple reconstruction algorithm, can be appliedto other composite solid-state spin systems, such as quantum dots and phosphorous donors in silicon. Such measurements could provide a powerful approach for the study of many-body dynamics of complex spin environments, potentially exhibiting nontrivial eects relatedto the interplay between nuclear and electronic spin baths. 124

142 Chapter 8 Outlook 8.1 Sensitivity of an NV magnetic sensor In addition to the techniques discussed in this dissertation, there have been a number of eorts in recent years to improve the sensitivity of NV based magnetometers. Recall from Chapter 2 that NV sensitivity is dependent on three main factorsmeasurement contrast α, number of photons collected per measurement β, and measurement time T (limited by the NV spin dephasing time T 2 for DC measurements and coherence time T 2 (n) for AC measurements)given by the following equation: η 1 α βt. (8.1) We summarize recent improvements to each of these factors individually. 125

143 Chapter 8: Outlook Measurement contrast In Chapter 5, we discussed using diamonds that have been CVD-grown such that the NV centers are preferentially oriented along two axes in order to increase the contrast of an ensemble measurement by a factor of 2. As discussed previously however, using these diamonds has several disadvantages. First, these samples only allow for the measurement of magnetic eld projections along two independent axes whereas three are necessary for a full vector reconstruction. Also, because preferential NV orientation has thus far only been observed in CVD-grown samples, there are constraints on the additional material processing that can be performed on the diamond. Further irradiation and annealing to increase the typically poor as-grown N-to-NV conversion will destroy the preferential orientation. However, a technique wherein an external strain is introduced to the diamond crystal during annealing in order to induce a re-orientation of the defects along a preferred direction has been demonstrated on neutral carbon <100>-split interstitial defects in diamond [89] and may also prove applicable to NV centers. There are other possible methods for improving measurement contrast, which may be employed in addition to or as an alternative to using samples with preferential NV orientation. For example, an additional MW π pulse may be applied to the three degenerate non-aligned NV orientation classes after initialization to transfer their spin population into the less uorescent 1 state, thus reducing their contributions to the background. Another technique for reducing background uorescence is to apply a high enough static eld ( 500 G) such that NV spin state mixing suppresses uorescence in the non-aligned NV orientation classes [130]. These additional contrast 126

144 Chapter 8: Outlook improvement techniques and others will be explored in future work Number of photons collected per measurement There are two main factors that determine the number of photons collected per measurement: the number of NV sensors probed in the measurement and the photon collection eciency of the experimental apparatus. Number of NV sensors As discussed previously, increasingthe number of NV sensors involved in the measurementi.e., the NV concentration in the measurement volumein practice often requires a correspondingincrease in N incorporation into the diamond. The increased N electronic spin impurity concentration decreases the NV spin coherence time which in turn adversely aects the magnetic sensitivity. Instead, much recent work has focused on improvingthe N-to-NV conversion eciency, which is typically very low in as-grown N-doped CVD samples (< 0.5%) and standard N ion implanted samples ( 1-5%). By irradiatingwith, e.g., electrons [36, 34, 35], protons [34], neutrons [33], carbon ions [17], or heavy ions [36] in order to create vacancies in the diamond crystal and subsequently annealing, N-to-NV conversion eciencies as high as 50% have been achieved [16, 17]. Photon collection eciency The refractive index of diamond is 2.42, which corresponds to a critical angle of 24.5 at a diamond-air interface and, in a typical bulk setup, 90% of the 127

145 Chapter 8: Outlook 4 mm Figure 8.1: Red-ltered photograph of NV uorescence from a planar diamond sample containing a relatively high NV concentration (NV 1 ppm). Guiding of NV uorescence light is evident as a bright glow around the diamond's perimeter, while a 532-nm laser beam passes through its center [70]. NV uorescence being totally internally reected back into the diamond. Taking into account the numerical aperture (NA)of the objective lens, the loss through various optical elements in the experimental apparatus, and the detection eciency of the CCD, a typical wide-eld setup collects 3-8% of the NV uorescence photons emitted during a measurement. In order to improve photon collection eciency, there have been various recent eorts to fabricate diamond microstructures designed to couple NV uorescence more strongly to a detector. For example, a diamond solid immersion lens (SIL)of radius 500 μm demonstrated a collection eciency of slightly less than 20% [67], whereas another diamond SIL of radius 2.5 μm gave a (calculated)collection eciency of 30% [68]. Another geometry that has been recently reported is a diamond nanowire; such a waveguiding microstructure of radius 100 nm and height 2 μm demonstrated a collection eciency of 40% [69]. An alternative to fabricating microstructures in diamond, which can be quite 128

146 Chapter 8: Outlook a dicult material to etch, is to simply collect the totally internally reected NV uorescence as it escapes out of the edges of a typical planar diamond sample (see Figure 8.1). A prototype device in which four photon detectors are positioned at the edges a diamond sample has demonstrated a collection eciency of 47%, with up to 90% theoretically achievable [70] Coherence Time In ultra-pure diamonds engineered with low N electronic spin impurities and low 13 C nuclear spin impurities, single NV center spin coherence times as long as 2 ms have been observed at room temperature [1]. By applying dynamical decoupling techniques as discussed in Chapter 6, comparable coherence times have been achieved in NV ensembles with higher NV concentrations desirable for enhanced magnetic sensitivity. The NV spin coherence time is limited by spin-phonon relaxation to several milliseconds at room temperature; however as discussed in Chapter 6, for magnetic sensing applications where the temperature requirements are less constrained, longer coherence times and hence more sensitive magnetic eld measurements may be achieved by operating at colder temperatures. An NV coherence time T (n) 2 10 ms is achievable with thermoelectric cooling and T (n) 2 > 0.5 s with liquid nitrogen cooling [131]. 8.2 Spatial resolution of an NV magnetic sensor One region of parameter space that NV based magnetic sensors can cover which is inaccessible to most currently available magnetic sensing technologies is in per- 129

147 Chapter 8: Outlook 10 2 Magnetic Sensitivity (T/ Hz) SQUIDs Hall Probes BECs MRFMs Vapor Cells NV Centers Spatial Resolution (m) Figure 8.2: Plot of the spatial resolution versus magnetic sensitivity of a variety of recently demonstrated magnetic sensing technologies: superconducting quantum interference devices (SQUIDs) [132, 133, 134, 135, 136, 137, 138, 139], scanning Hall probes [140, 141, 142, 143, 144], magnetometers employing Bose-Einstein condensates (BECs) [145, 146], magnetic resonance force microscopes (MRFMs) [147, 148], atomic vapor cells and spin exchange relaxation-free (SERF) magnetometers [149, 150, 151, 152], and NV magnetometers [65, 59, 85, 153, 70, 154, 155]. forming sensitive magnetic eld measurements with high spatial resolution under ambient conditions. Figure 8.2 plots the spatial resolution and magnetic sensitivity of a variety of recently demonstrated magnetometers. Note that NV based magnetic sensors have the added benet of operating sensitively under ambient conditions. In contrast, superconducting quantum interference devices (SQUIDs) require cryogenic temperatures; Bose-Einstein condensates (BECs) require vacuum; and magnetic resonance force microscopes (MRFMs) operate optimally under vacuum at cryogenic temperatures. Furthermore, while the wide-eld and confocal setups discussed in 130

148 Chapter 8: Outlook this dissertation are fundamentally limited by optical diraction to 300 nm, several techniques have been proposed and demonstrated to achieve signicantly better spatial resolution. Scanning tip magnetometer There have been several recent demonstrations of magnetic sensing using single NV centers in a scanning-tip geometry. Rondin et al. attached a diamond nanocrystal containing a single NV to an atomic force microscope (AFM) tip and performed DC magnetometry with 9 μt/ Hz sensitivity and tens of nm spatial resolution [154]. Maletinsky et al. fabricated a diamond nanopillar containing a single NV center at the tip and demonstrated AC magnetometry with 56 nt/ Hz sensitivity and 10 nm spatial resolution [155]. These initial demonstrations already surpass the spatial resolutions achieved by any of the alternative magnetic sensing technologies except for MRFM, which has the same geometry. It is important to note however that while the scanning-tip geometry allows for signicantly improved spatial resolution, there are also trade-os: due to the small volume of the tip, incorporating more than one NV center into the desired region is rather dicult without signicantly degrading the NV spin coherence and consequently magnetic sensitivity. The NV center often also experiences surface eects which can shorten the coherence time. We will discuss these trade-os in more detail in the next section. Stimulated emission depletion microscopy Another approach for achieving sub-optical-diraction-limited spatial resolution is to employ super-resolution optical schemes. For example, Rittweger et al. were 131

149 Chapter 8: Outlook able to use a stimulated emission depletion (STED) microscope to resolve single NV centers with 5.8 nm spatial resolution [156]. This technique has several advantages over a scanning tip geometry in that diamond fabrication is not necessary and higher NV concentrations are more easily probed. Thus far, simultaneous STED microscopy and DC magnetometry has achieved 1 μt/ Hz sensitivity and 30 nm spatial resolution [153], with further improvements in progress. 8.3 Discussion of trade-os One of the most powerful advantages of NV based magnetic sensors is their versatility. In this dissertation, we have discussed a variety of diamond geometries (e.g., NV centers homogeneously distributed throughout a bulk sample, NV centers occupying a thin layer at the surface of a bulk sample, and NV centers at the tip of a scanning probe); measurement schemes (e.g., side-collection, simultaneous wide-eld imaging onto a CCD, scanning confocal microscopy, and STED microscopy); and techniques for improving magnetic sensitivity (e.g., preferential NV orientation, dynamical decoupling, irradiation and annealing, and performing measurements at low temperatures). However as discussed previously, each technique has advantages and disadvantages. A CCD based measurement scheme allows for simultaneous imaging of magnetic elds over large regions, but spatial resolution is limited by optical diraction. A scanning tip geometry and STED microscopy allow for sub-optical-diraction spatial resolution, but both require scanning and are consequently limited in how rapidly large elds of view may be imaged. Also, while many of the geometries, measurement schemes, and magnetic sensitivity enhancing techniques can be combined, 132

150 Chapter 8: Outlook some are incompatible. For example, irradiating and annealing diamond samples in order to increase NV concentration and subsequently improve magnetic sensitivity currently irreversibly destroys any preferential NV orientation and the corresponding enhancements to measurement contrast and magnetic sensitivity. Fortunately, the high degree of exibility in NV based magnetic sensor implementation allows for the optimization of dierent NV magnetometers for dierent applications. 8.4 Applications The ability to map magnetic elds with high sensitivity and spatial resolution at room and near-room temperature may provide additional information in a variety of areas that were previously inaccessible with current magnetic sensing technologies. In particular, an NV based magnetic imager may be applied to study bio-magnetic elds in living organisms, recently demonstrated for the rst time on magnetotactic bacteria (see Figure 8.3) [54]. Another potential bio-application for an NV ensemble magnetic eld imager is in studying connectivity and signaling in functional, cultured neuronal networks. These collections of neurons interconnected by synapses provide the physical basis of central and peripheral nervous systems, but despite rapid advances in neuroscience, a clear understanding has yet to emerge of how microscopic connectivity of neurons encodes macroscopic function of the network. Determining the rules for translating neuronal connectivity to network function will require experimental tools that are capable of real-time mapping of both local and global spatiotemporal connectivity among neurons and signal propagation between them. However, given the size of a biologically 133

151 Chapter 8: Outlook 0.2 a b c 5 μm d 5 μm 10 μm 10 μm Magnetic field projection (G) Magnetic field projection (G) Figure 8.3: Wide-eld optical and magnetic images of magnetotactic bacteria. (a) Bright-eld optical images of MTB adhered to the diamond surface while immersed in phosphate buer solution. Several out-of-focus bacteria are also visible oating above the diamond surface. (b) Image of magnetic eld projection along the [111] crystallographic axis in the diamond for the same region shown in (a) determined from NV ESR spectra. Outlines are superimposed on the magnetic eld image to indicate the locations of the MTB determined from bright-eld images, and the colour of the outlines indicate the results of a bacterial live-dead assay performed after measuring the magnetic eld (black for living, red for dead, and grey for indeterminate). (c) Bright-eld image of several dried MTB on the surface of the diamond chip. (d) Image of magnetic eld projection along the [111] axis for the same region shown in (c) determined from NV ESR spectra, with outlines to indicate the locations of MTB determined from (c) [54]. 134

152 Chapter 8: Outlook relevant, functional neuronal network ( 1 mm or larger), such real-time functional imaging is a formidable challenge beyond the capabilities of existing experimental instruments and methodologies. With further technical improvements such as those discussed in this dissertation, an optimized NVensemble magnetic eld imager may be able to address this challenge by providing micron-scale mapping of magnetic elds on a sub-millisecond time scale for all neurons within a functional network cultured on the bio-inert surface of a diamond magnetometer chip. This mapping of neuronal activity via magnetic eld imaging is minimally invasive and can be correlated against stimulation and/or reward administration to identify the presence and emergence of persistent patterns of local and global activity. NVmagnetic sensors have also recently been employed to measure nuclear magnetic resonance from organic material deposited at the surface of the diamond substrate [55, 56]. More potential applicationssuch as in biomagnetism (e.g., cardiac cells, magnetic organs used for navigation, etc.), novel material characterization (e.g., graphene), and paleomagnetism (e.g., dynamo theory)will be explored as the capabilities NVmagnetic sensing technology continue to improve. 135

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154 Appendix A NV Magnetic Sensitivity A.1 DC magnetometry: ESR To derive the optimum sensitivity of an NV magnetometer which uses ESR spectra to measure DC magnetic elds, rst consider the intensity prole of an NV resonance in the ESR spectrum (see Figure A.1): I(ν m )=R [ 1 αf ( )] νm ν 0 Δν (A.1) where R is the rate of detected photons per second, α is the contrast of the resonance dip, F is a function describing the lineshape of the resonance, Δν is the linewidth (full-width at half-maximum) of the resonance, and ν 0 is the resonance frequency. Recall that the resonance frequency is related to the magnetic eld by the simple Larmor equation ν 0 = γb 0, where B 0 is the projection of the magnetic eld along the NV symmetry axis and the NV gyromagnetic ratio is γ = gμ B h 2.8 MHz/G. The linewidth Δν is fundamentally limited by the characteristic dephasing time T 2, which is dependent on inhomogeneities in the magnetic environment of the NV 137

155 Appendix A: NV Magnetic Sensitivity ν 0 R Intensity I δi Δν αr δν m MW Frequency ν m Figure A.1: Intensity prole of an NV resonance in the ESR spectrum illustrating the sensitivity of an ESR based DC magnetic eld measurement. center(s), as discussed in Chapter 1. The lineshape F of the resonance may be described by a Gaussian function (e.g., when a large number of nuclear 13 C spins dominate the dephasing), a Lorentzian function (e.g., when paramagnetic spins corresponding to nitrogen defects dominate the dephasing or when the resonance is power-broadened due to continuous laser or microwave excitation), or possibly a convolution of both (i.e., a Voigt function). The uorescence intensity I is most sensitive to small changes in the magnetic eld at the point of maximum slope: max I ν m = αr ΔνP F (A.2) We dene the numerical parameter P F, which is related to the specic prole F of NV resonance. For a Gaussian lineshape, P F = e/8ln2 0.70; for a Lorentzian lineshape, P F =4/ [57]. Consider a measurement of duration t m. Assuming low contrast α, the number of 138

156 Appendix A: NV Magnetic Sensitivity photons collected per measurement β = It m can be approximated by β Rt m, and the corresponding photon shot noise can be approximated by δβ Rt m. The shotnoise-limited minimum detectable eld δb min for this measurement can be calculated from the photon shot noise at the point of maximum slope (Figure A.1): γδb min = δν m,min = δβ max νm β Rtm t m max νm I (A.3) Substituting in Equation A.2, Equation A.3 becomes: h Δν δb min P F gμ B α β (A.4) The sensitivity η esr of the measurement is then given by the following relation [60, 52]: tm h Δν η esr = δb min tm = P F gμ B α β (A.5) The magnetic eld sensitivity is dependent on resonance contrast α, number of photons collected per measurement β, resonance linewidth Δν, and measurement duration t m. These parameters are not independent of each other: the resonance contrast α may be increased by increasing the power of the microwave excitation at the expense of increasing the linewidth Δν due to power broadening. Likewise, the linewidth Δν may be decreased (up to a limit dened by the NV spin characteristic dephasing time T 2 ) by lowering the laser excitation power to reduce power broadening; however doing so decreases number of photons collected per measurement β. Optimizing these parameters to achieve best DC magnetic eld sensitivity using the ESR method yields [57]: 2 1 η esr P F gμ B α βt 2 (A.6) 139

157 Appendix A: NV Magnetic Sensitivity A.2 DC magnetometry: Ramsey sequence To derive the optimum sensitivity of an NV magnetometer that uses the Ramsey pulse sequence to measure DC magnetic elds, rst consider the signal S, dened previously as the number of photons collected in one measurement and dependent on the magnitude B of the static magnetic eld to be measured. (Note that B is an additional static eld separate from the known static eld B 0, which was applied to spectrally distinguish the four NV orientation classes.) As a result, the unknown magnetic eld magnitude B can be extracted from the signal S, and the uncertainty in the signal δs is related to a corresponding uncertainty in the measured eld δb by the slope: δs = S B δb (A.7) The signal is most sensitive to small changes in the magnetic eld at the point of maximum slope. As a result, the minimum resolvable magnetic eld δb min is given by: δb min = δs max S B (A.8) where the maximum slope can be determined from the Ramsey magnetometry curve given in the main text by Equation 2.3: max S B = max 2πγτ (a b) 2 sin (2πγBτ) =2πγτ (a b) 2 (A.9) Equation A.9 can be written in terms of the average number of photons collected per measurement β =(a + b)/2 and the measurement contrast α =(a b)/(a + b): max S B (a b) (a + b) =2πγτ (a + b) 2 =2πγταβ (A.10) 140

158 Appendix A: NV Magnetic Sensitivity a τ = 0.49 μs b Fluorescence Signal S Fluorescence Signal S δs δb a b Free Precession Time τ (μs) Static Magnetic Field B (G) Figure A.2: Sensitivity of a Ramsey based DC magnetic eld measurement. (a) Modeled Ramsey FID curve, with the free precession time τ =0.49 μs used for the magnetic measurement indicated by a red arrow. (b) Example magnetometry curve illustrating the uncertainty δb in the measured magnetic eld due to uctuations δs in the uorescence signal. In typical NV measurements, the number of photons collected from each NV is quite small. As a result, the uncertainty in the signal is dominated by photon shot noise and can be approximated by δs β: δb min β 2πγταβ 1 gμ B τ 1 α β (A.11) In more advanced measurement schemes with higher photon collection and measurement contrast (e.g., single-shot read-out), spin projection noise must also be incorporated into the uncertainty δs [52]. Note that other noise sources, such as electronic noise, may also limit the magnetic sensitivity. The sensitivity η ramsey of the measurement is then given by the following relation: η ramsey = δb min tm 1 1 τ gμ B α β (A.12) where the time t m of a measurement is approximated by the free precession time τ. As in the DC magnetometry scheme utilizing ESR spectroscopy, the magnetic eld sensitivity is dependent on resonance contrast α, number of photons collected 141

159 Appendix A: NV Magnetic Sensitivity per measurement β, and measurement duration τ. Also as before, these parameters are not independent. The measurement contrast decays as a result of NV spin dephasing, with characteristic time scale T 2. Thus, increasing the measurement duration τ degrades the contrast α; the optimum magnetic eld sensitivity is achieved when τ T 2: η ramsey 1 gμ B α βt 2 (A.13) A.3 AC magnetometry In this section, we derive the sensitivity of an NV magnetometer which employs a spin echo pulse sequence to measure an AC magnetic eld along the NV axis, described by b(t) =B sin (2πf ac t ϕ 0 ). The MW echo pulse of the sequence occurs at t = τ/2, where the full pulse duration τ is related to the magnetic eld frequency f ac as stated in Chapter 2. The phase φ accumulated by the NV spins over the pulse sequence duration is given by: φ = τ 2 0 φ =4πγB 2πγb(t)dt τ 2 0 τ τ 2 2πγb(t)dt sin (2πf ac t ϕ 0 )dt φ =4γBτ cos (ϕ 0 ) (A.14) (A.15) (A.16) For simplicity, let us focus on the optimal situation where the MW pulses coincide with the nodes of the AC eld [see Figure 2.3(a)]. In this case, ϕ 0 =0and the phase accumulated by the NV spins and subsequently the sensitivity to magnetic elds is maximized. As mentioned previously, AC magnetic elds with randomly uctuating 142

160 Appendix A: NV Magnetic Sensitivity a τ = μs b Fluorescence Signal S Fluorescence Signal S δs δb a b Spin Echo Time τ (μs) AC Magnetic Field Amplitude B (nt) Figure A.3: Sensitivity of a spin echo based AC magnetic eld measurement. (a) Measured spin echo decoherence curve of an NV ensemble, with the spin echo time τ = 324.4μs used for the magnetic measurement indicated by a red arrow. (b) Example magnetometry curve illustrating the uncertainty δb in the measured magnetic eld due to uctuations δs in the uorescence signal. phases may also be measured using a variant of the spin echo pulse sequence; however, the sensitivity is poorer, as discussed elsewhere [63, 64]. As in the Ramsey DC magnetometry technique, we dene the signal S as the number of photons collected in one measurement, where S oscillates between a (corresponding to a phase accumulation of φ = 0) and b (corresponding to a phase accumulation of φ = π): S = S = (a + b) 2 (a + b) (a b) 2 (a b) 2 cos (φ) cos (4γBτ) (A.17) (A.18) The unknown magnetic eld amplitude B can be extracted from the signal S, and the uncertainty in the signal δs is related to a corresponding uncertainty in the measured eld δb by the slope: δs = S B δb (A.19) 143

161 Appendix A: NV Magnetic Sensitivity The signal is most sensitive to small changes in the magnetic eld at the point of maximum slope, which from Equation A.18 is: max S B = max 4γτ max S B (a b) 2 =4γτ (a b) 2 sin (4γBτ) (A.20) (A.21) Equation A.21 can be written in terms of the average number of photons collected per measurement β =(a + b)/2 and the measurement contrast α =(a b)/(a + b): max S B (a b) (a + b) =4γτ (a + b) 2 =4γταβ (A.22) In typical NV measurements, the number of photons collected from each NV is quite small. As a result, the uncertainty in the signal is dominated by photon shot noise and can be approximated by δs β. The minimum resolvable magnetic eld δb min is therefore given by: δb min δb min = β 4γταβ δs max S B π 1 2gμ B τ 1 α β (A.23) (A.24) In more advanced measurement schemes with higher photon collection and measurement contrast (e.g., single-shot read-out), spin projection noise must also be incorporated into the uncertainty δs [52]. Note that other noise sources, such as electronic noise, may also limit the magnetic sensitivity. The sensitivity η se of the spin echo measurement is then given by the following relation: η se = δb min tm π 1 1 τ 2gμ B α β (A.25) where the measurement time t m is approximated by the full free precession time τ. 144

162 Appendix A: NV Magnetic Sensitivity Similar to the other NV magnetometry schemes discussed thus far, the magnetic eld sensitivity is dependent on resonance contrast α, number of photons collected per measurement β, and measurement duration τ. Again, these parameters are not independent; the measurement contrast decays as a result of NV spin decoherence, with characteristic time scale T 2. Thus, increasing the measurement duration τ degrades the contrast α; the optimum magnetic eld sensitivity for an spin echo measurement is achieved when τ T 2 [52]: η se π 1 2gμ B α βt 2 (A.26) As expected, the magnetic eld sensitivity of a spin echo based AC measurement is limited by T 2 rather than T 2 as in the Ramsey based DC measurement. Typical spin echo coherence times in high-purity samples are on the order of 500 μs, though T 2 times on the order of 2 ms have been observed in high-purity isotopically engineered samples [1]. 145

163 Appendix B Experimental Apparatus B.1 CCD based wide-eld uorescence microscope NV ensemble measurements were performed using a custom-built wide-eld uorescence microscope (see Figure B.1), a setup which has been used both for averaging the collective behavior of a large number of NV centers over a several-cubic-micron detection volume as well as for imaging NV behavior over large elds of view. In particular, the latter modality allowed for the demonstrations of 2-dimensional magnetic eld imaging with thin-layer NV-diamond samples discussed in Chapter 4. Optical excitation was provided by a 3-W 532-nm laser (Laser Quantum Opus) which illuminated a region of interest on the diamond surface through a 20 NA = 0.75 objective (Nikon). An acousto-optic modulator (AOM) (Isomet M1133-aQ80L- H) acted as an optical switch, to pulse the laser with precise timing in order to prepare and detect the NV spin states. A loop antenna was positioned near the diamond surface and connected to the amplied (Mini-circuits, ZHL-16W-43-S+) output of a 146

164 Appendix B: Experimental Apparatus a b LASER Laser Quantum Opus HWP f = 100 mm f = 300 mm f = 50 mm CCD Starlight Xpress SXV-H9 bandpass filters 640-nm longpass 750-nm shortpass f = 300 mm iris AOM f = 50 mm f = 300 mm HWP defocusing lens diamond permanent magnet f = 50 mm dichroic mirror Semrock FF552-Di02 objective Nikon MW loop optical chopper New Focus 3501 Legend lens mirror Isomet M1133-aQ80L-H Figure B.1: CCD based wide-eld uorescence microscope (a) photograph and (b) schematic. 147

165 Appendix B: Experimental Apparatus microwave signal generator (Agilent E8257D) to generate a homogeneous MW eld over the region of interest. Fast switching of the microwave eld (ZASWA-2-50DR) enabled coherent manipulation of the NV spin states necessary for applying standard pulse sequences, such as spin echo and CPMG. Phase modulation of the MW pulses was achieved using an in-phase/quadrature (IQ) mixer (Marki IQ-1545). Optical and MW pulse timings were controlled through a computer-based digital delay generator (SpinCore PulseBlaster PRO ESR500). NV uorescence was collected by the objective, ltered through a dichroic mirror (Semrock LM ) and additional lters, and imaged onto a cooled Charge- Coupled Device (CCD) camera (Starlight Xpress SXV-H9). As the duration of a single measurement is shorter than the minimum exposure time of the camera, the measurement was repeated for several thousand averages within a single exposure and synchronized to an optical chopper (New Focus 3501) placed before the CCD in order to block uorescence from the optical initialization pulse. B.2 Scanning confocal microscope NV measurements on single centers were performed using a custom-built scanning confocal microscope. Optical excitation was provided by a 300-mW 532-nm diode pumped solid-state (DPSS) laser (Changchun New Industries Optoelectronics Tech MLLIII ), focused onto the sample using a 100, NA = 1.3 oil immersion objective (Nikon CFI Plan Fluor 100x oil). The excitation laser was pulsed by focusing it through an AOM (Isomet 1205C-2). NV uorescence was collected through the same objective and separated from the excitation beam using a dichroic lter (Sem- 148

166 Appendix B: Experimental Apparatus a b f = 250 mm iris Legend iris AOM linear polarizer aspheric lens HWP lens mirror fiber coupler f = 150 mm SM PM fiber iris f = 250 mm 532-nm notch filter LASER f = 150 mm objective Nikon 100x diamond HWP dichroic mirror f = 250 mm filter nm objective Olympus 10x linear polarizer f = 250 mm SM fiber objective Olympus 10x SPCM Figure B.2: schematic. NV scanning confocal microscope (a) photograph and (b) 149

167 Appendix B: Experimental Apparatus rock LM ). The light was additionally ltered (Semrock LP02-633RS-25) and focused onto a single-photon counting module (Perkin-Elmer SPCM-ARQH-12). MW was delivered to the sample using a 20 μm-diameter wire soldered across the diamond. The wire was driven by an amplied (Mini-circuits ZHL-16W-43-S+) MW synthesizer (Agilent E4428C). Phase modulation of the MW pulses was achieved using an in-phase/quadrature (IQ) mixer (Marki IQ-1545). Microwave and optical pulses were controlled using a computer-based digital delay generator (SpinCore PulseBlaster ESR400). B.3 Low temperature setup For NV measurements at low temperatures, the sample was mounted inside a continuous-ow cryostat (Janis ST-500) with active temperature control (Lakeshore 331). Optical excitation was applied with a 532-nm laser beam focused on the diamond surface through a NA = 0.6 air objective. The resulting NV uorescence was collected through the same objective and directed to a multimode ber coupled to a single-photon counting module (Perkin-Elmer SPCM-ARQH-12). This semi-confocal experimental setup resulted in a 30 μm 3 detection volume. Microwave control pulses were delivered to the NV spins using a 70 μm-diameter copper wire at the diamond surface. The wire was driven by an amplied (Mini-circuits ZHL-16W-43- S+) MW synthesizer (Stanford Research Systems SG 384). Phase modulation of the MW pulses was achieved using an in-phase/quadrature (IQ) mixer (Marki IQ-1545). Microwave and optical pulses were controlled using a computer-based digital delay generator (SpinCore PulseBlaster ESR400). 150

168 Appendix B: Experimental Apparatus Figure B.3: Photograph of the low temperature NV measurement setup. 151